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The magnetic field dependence of the phonon resistivity of ErVO4 in the paramagnetic phase
Volume 73A, number 5,6
PHYSICS LErrERS
15 October 1979
THE MAGNETIC FIELD DEPENDENCE OF THE PHONON RESISTWITY OF ErVO4 IN THE PARAMAGNETIC PHASE V.P. SRIVASTAVA and G.S. VERMA Physics Department, Banaras Hindu University, Varanasi-221005, India Received 8 February 1979 Revised manuscript received 30 July 1979
An expression for the spin—phonon relaxation rate for a spin-I /2 Kramer salt has been used to explain the magnetic field dependence of the phonon resistivity of ErVO4 in the paramagnetic phase for fields upto 24 kOe. A gaussian line shape function reproduces the main features of the experimental result. 3~ground state is a 4115/2 level which is splitThe intoErKramer’s doublets in a crystal field of tetrag-
onal symmetry. Spectroscopic investigations have revealed a ground doublet withg valuesg~= 3.85 ±0.3 and g~= 8.5 ±0.5 separated from two higher doublets by 43.6 cm—’ and 45.4 cm—1, respectively. At low temperatures, only the ground state doublet is operative in the resonant scattering of phonons. The system in the presence of a magnetic field is thus effectively equivalent to a spin-i /2 system. The relaxation rate for this type of resonant scattering is given by [11 2 r~
Taking g(11w 1~w
2/~2]
—
0)= (1 [A.../~) exp [—(11w 1iw0) —
(2) where ~ is the effective gaussian width for the transition causing resonant scattering, the expression for the relaxation rate becomes r~(x, H, T) = GxTtanh (pH/2T)
(3) X exp [—(x
—
pH/T)2/(/.~ti/kT)2]
where
=-~--IEIfIEV,~’Ii>~ Mu2Lm n ]g(hw_hwo)A.No~ (1)
where Ii), If) are the initial and final spin states, respectively. V,~is a phenomenological spin—phonon (orbit— lattice) coupling constant, g(hw 17w 0) is the homogeneous spin—phonon line shape function, 11w0 is the energy difference between the spin levels Ii> and If), v is the phonon (i.e. sound) velocity and i.~N0is the difference in spin population between the levels Ii) and If). Mis the mass of the specimen. ,~N0in the present case is found to be N0 tanh (pH/2T), where N0 is the total number of spins in the crystal and p = gil/k (g, j3, k having their usual meaning), We have tried both gaussian and lorentzian line shapes for g(hw lrw0) but the former is found to give better agreement with the experiment results.
~./~kN G
2 07 Mi.~Mu2
11w
x 72
=
E m En
The magnetic field dependence of the thermal re-
—
—
sistivity of ErVO4 has been studied by Metcalfe and Rosenberg [2] in the temperature range 0.1 to 1.2K and in magnetic fields up to 30 kOe. Spectroscopic measurements show that it remains paramagnetic at a temperature at least as low as 0.5 K. Metcalfe and Rosenberg observed that on applying a magnetic field, the termal resistivity in the paramagnetic phase increases with magnetic field, passes through a broad maximum and then decreases to a value near that in zero field. Moreover, this maximum moves to higher fields as the temperature is raised. These features can be reasonably well explained by expression (2). 433
Volume 73A, number 5,6
PHYSICS LETTERS
15 October 1979
Fig. 1 shows the variation of the thermal resistivity with magnetic field calculated in the framework of Callaway’s theory of lattice thermal conductivity [3] using eq. (3) along with the experimental results of ref. [21. In fig. ld, the value of g = 3.85 as found by
360
be seen that the theoretical peaks are shifted towards ~ 280 E 320 240 U
higher field values. This is truehave at other temperatures too. Metcalfe and Rosenberg also noted that g values spectroscopic known measurements from other experiments has been used cannot butexplain it can
— —
,,,/~
10722
their place results. withg peaks as seen are the reproduced in figs. la,b,c. theline In fig. right ld, the curves 1, =The 2 7.44 and 3 show effect ofatthe width
200
on the peak — as the width is increased, the flatness increases. The best fit is obtained for i~= 0.90 K. In
60
20
fig. 1 c G and L show the theoretical curves with a of the form (2) and a lorentzian of the form 2 but at all temperatures it is 2T found 2/ir[lthat + lj(w the value — w0)] of the resistivity even at maximum applied field does not approach its zero field value in the case of the lorentzian while it attains its
gaussian 0
4
8
2 16 H (KOe)
20
240
=
200 E U
60
wings give some contribution to the resistivity even at high fields. The fact that the phonon absorption line is more zero field likevalue a gaussian using the than gaussian, a lorentzian i.e., the indicates lorentzian that normal spin—spin interaction is also important, which
,,,/~~~TO86K~86K
120 80
0
4
8
80
12 16 H (KOe)
24
1
E 140
o
4
8
180
lIl°K
L 12 16 H (KOe) (d)
20
24
whilethe kOe thesplitting nearest is next onlydoublet 10.38 is cm—’ at 43.6 (with cm—’. g = 7.44) Fig. 2 shows the physical process responsible for
24
Fig. 1. The thermal resistivity, W, against the applied field H, for c-axis II Q II H at different temperatures. Solid curves represent the theoretical result, curves with circles represent the experimental result of ref. [2]. In (d) for T 1.11 K, curves 1, 2 and 3 correspond to three different values of ~, viz., 0.90, 1.2 and 1.8 K, respectively. In (c), G and L refer to the theoretical results with gaussian and lorentzian line shape
the nature of the curve observed. The shaded area represents the band of phonons “missing” from the pho-
1= I•II 1<
140
=
100
60 0
field. We have looked for an explanation of this feature in terms of coupled spin—phonon modes but af-
ter investigation, it is found that such modes cannot give rise to this feature. There is also no possibility of mixing of ground state doublet levels with higher levels as the field is increased because even at H = 30
(c)
U
E U
20
has not been taken into account in the present model. This may be one of the reasons for the poor agreement between theory and experiment at very high magnetic
4
8
12
16
H (KOe)
20
functions, respectively.
434
PHYSICS LETTERS
Volume 73A, number 5,6
IO-= f
u
9
;
8
c:
7
3
15 October 1979
10-6
-2 9 3
28
.r=7 $6
z L 8 2 5 E 0, 4 ; 0 3 2 : = id7
z 5 : %4 O3 z o 2 I 012345878 X-%W/(kT)
“-6
I
2
3
4
5
8
7
X=hd(kT)
8
012345678 X=ho/
(kT)
Fig. 2. Calculated heat current plotted as a function of x =Rw/kT for three different magnetic fields at T = 1 .ll K. The shaded area represents the heat current removed from the phonon spectrum. (a) At H = 2 kOe, a small number of phonons are removed. (b) At H = 6 kOe, a maximum number of phonons are removed. (c) At H = 10 kOe, the number of phonons removed decreases.
non spectrum. As the field parallel to the hexagonal axis is increased from zero, one band of phonons is removed from the phonon spectrum corresponding to the transition between ground state doublet levels and the thermal resistivity increases. The number of phonons removed increases with magnetic field. A maximum number of phonons are removed at 6 kOe and thereafter the number decreases causing the thermal resistivity to increase first, pass through a maximum at 6 kOe and then decrease. As the field is increased beyond 12 kOe, there are very few phonons available for scattering and the resistivity approaches its zero field value. For the present system, the spin-phonon coupling constant is found to be 7 = 11.80 cm-l and the line width A = 0.63 cm-l. Since there has been little quantitative work on ErVO,, we are not able to identify the different sources contributing to the line width nor to get an estimate of 7. But it is worth mentioning that they are of the same order as that found by Metcalfe and Rosenberg [4,5] for cerium~ethylsulphate
and holmium ethylsulphate, respectively. The value of the point defect scattering parameter A (A-l = Aw4) is found to be A = 1.42 X 10m46 s4 cm-l which is also of the same order as that found in ref. [4] for CES. The values of the other parameters used in the calculation are u = 5 X lo5 cm/s, the Casimir length L = 0.1384 cm. One of us (VI’S) is indebted to C.S.I.R., India, for financial support. References [l] R. Orbach, Phys. Rev. Lett. 8 (1962) 393. [2] M.J. Metcalfe and H.M.Rosenberg, J. Phys. C5 (1972) 474. [3] J. Callaway, Phys. Rev. 113 (1959) 1046; 122 (1961) 787. [4] P.V.E. McClintock, I.P. Morton, R. Orbachand H.M. Rosenberg, Proc. Roy. Sot. A298 (1967) 359. [5] P.V.E. McClintock and H.M. Rosenberg, Proc. Roy. Sot. A302 (1968) 419.
435
PHYSICS LErrERS
15 October 1979
THE MAGNETIC FIELD DEPENDENCE OF THE PHONON RESISTWITY OF ErVO4 IN THE PARAMAGNETIC PHASE V.P. SRIVASTAVA and G.S. VERMA Physics Department, Banaras Hindu University, Varanasi-221005, India Received 8 February 1979 Revised manuscript received 30 July 1979
An expression for the spin—phonon relaxation rate for a spin-I /2 Kramer salt has been used to explain the magnetic field dependence of the phonon resistivity of ErVO4 in the paramagnetic phase for fields upto 24 kOe. A gaussian line shape function reproduces the main features of the experimental result. 3~ground state is a 4115/2 level which is splitThe intoErKramer’s doublets in a crystal field of tetrag-
onal symmetry. Spectroscopic investigations have revealed a ground doublet withg valuesg~= 3.85 ±0.3 and g~= 8.5 ±0.5 separated from two higher doublets by 43.6 cm—’ and 45.4 cm—1, respectively. At low temperatures, only the ground state doublet is operative in the resonant scattering of phonons. The system in the presence of a magnetic field is thus effectively equivalent to a spin-i /2 system. The relaxation rate for this type of resonant scattering is given by [11 2 r~
Taking g(11w 1~w
2/~2]
—
0)= (1 [A.../~) exp [—(11w 1iw0) —
(2) where ~ is the effective gaussian width for the transition causing resonant scattering, the expression for the relaxation rate becomes r~(x, H, T) = GxTtanh (pH/2T)
(3) X exp [—(x
—
pH/T)2/(/.~ti/kT)2]
where
=-~--IEIfIEV,~’Ii>~ Mu2Lm n ]g(hw_hwo)A.No~ (1)
where Ii), If) are the initial and final spin states, respectively. V,~is a phenomenological spin—phonon (orbit— lattice) coupling constant, g(hw 17w 0) is the homogeneous spin—phonon line shape function, 11w0 is the energy difference between the spin levels Ii> and If), v is the phonon (i.e. sound) velocity and i.~N0is the difference in spin population between the levels Ii) and If). Mis the mass of the specimen. ,~N0in the present case is found to be N0 tanh (pH/2T), where N0 is the total number of spins in the crystal and p = gil/k (g, j3, k having their usual meaning), We have tried both gaussian and lorentzian line shapes for g(hw lrw0) but the former is found to give better agreement with the experiment results.
~./~kN G
2 07 Mi.~Mu2
11w
x 72
=
E m En
The magnetic field dependence of the thermal re-
—
—
sistivity of ErVO4 has been studied by Metcalfe and Rosenberg [2] in the temperature range 0.1 to 1.2K and in magnetic fields up to 30 kOe. Spectroscopic measurements show that it remains paramagnetic at a temperature at least as low as 0.5 K. Metcalfe and Rosenberg observed that on applying a magnetic field, the termal resistivity in the paramagnetic phase increases with magnetic field, passes through a broad maximum and then decreases to a value near that in zero field. Moreover, this maximum moves to higher fields as the temperature is raised. These features can be reasonably well explained by expression (2). 433
Volume 73A, number 5,6
PHYSICS LETTERS
15 October 1979
Fig. 1 shows the variation of the thermal resistivity with magnetic field calculated in the framework of Callaway’s theory of lattice thermal conductivity [3] using eq. (3) along with the experimental results of ref. [21. In fig. ld, the value of g = 3.85 as found by
360
be seen that the theoretical peaks are shifted towards ~ 280 E 320 240 U
higher field values. This is truehave at other temperatures too. Metcalfe and Rosenberg also noted that g values spectroscopic known measurements from other experiments has been used cannot butexplain it can
— —
,,,/~
10722
their place results. withg peaks as seen are the reproduced in figs. la,b,c. theline In fig. right ld, the curves 1, =The 2 7.44 and 3 show effect ofatthe width
200
on the peak — as the width is increased, the flatness increases. The best fit is obtained for i~= 0.90 K. In
60
20
fig. 1 c G and L show the theoretical curves with a of the form (2) and a lorentzian of the form 2 but at all temperatures it is 2T found 2/ir[lthat + lj(w the value — w0)] of the resistivity even at maximum applied field does not approach its zero field value in the case of the lorentzian while it attains its
gaussian 0
4
8
2 16 H (KOe)
20
240
=
200 E U
60
wings give some contribution to the resistivity even at high fields. The fact that the phonon absorption line is more zero field likevalue a gaussian using the than gaussian, a lorentzian i.e., the indicates lorentzian that normal spin—spin interaction is also important, which
,,,/~~~TO86K~86K
120 80
0
4
8
80
12 16 H (KOe)
24
1
E 140
o
4
8
180
lIl°K
L 12 16 H (KOe) (d)
20
24
whilethe kOe thesplitting nearest is next onlydoublet 10.38 is cm—’ at 43.6 (with cm—’. g = 7.44) Fig. 2 shows the physical process responsible for
24
Fig. 1. The thermal resistivity, W, against the applied field H, for c-axis II Q II H at different temperatures. Solid curves represent the theoretical result, curves with circles represent the experimental result of ref. [2]. In (d) for T 1.11 K, curves 1, 2 and 3 correspond to three different values of ~, viz., 0.90, 1.2 and 1.8 K, respectively. In (c), G and L refer to the theoretical results with gaussian and lorentzian line shape
the nature of the curve observed. The shaded area represents the band of phonons “missing” from the pho-
1= I•II 1<
140
=
100
60 0
field. We have looked for an explanation of this feature in terms of coupled spin—phonon modes but af-
ter investigation, it is found that such modes cannot give rise to this feature. There is also no possibility of mixing of ground state doublet levels with higher levels as the field is increased because even at H = 30
(c)
U
E U
20
has not been taken into account in the present model. This may be one of the reasons for the poor agreement between theory and experiment at very high magnetic
4
8
12
16
H (KOe)
20
functions, respectively.
434
PHYSICS LETTERS
Volume 73A, number 5,6
IO-= f
u
9
;
8
c:
7
3
15 October 1979
10-6
-2 9 3
28
.r=7 $6
z L 8 2 5 E 0, 4 ; 0 3 2 : = id7
z 5 : %4 O3 z o 2 I 012345878 X-%W/(kT)
“-6
I
2
3
4
5
8
7
X=hd(kT)
8
012345678 X=ho/
(kT)
Fig. 2. Calculated heat current plotted as a function of x =Rw/kT for three different magnetic fields at T = 1 .ll K. The shaded area represents the heat current removed from the phonon spectrum. (a) At H = 2 kOe, a small number of phonons are removed. (b) At H = 6 kOe, a maximum number of phonons are removed. (c) At H = 10 kOe, the number of phonons removed decreases.
non spectrum. As the field parallel to the hexagonal axis is increased from zero, one band of phonons is removed from the phonon spectrum corresponding to the transition between ground state doublet levels and the thermal resistivity increases. The number of phonons removed increases with magnetic field. A maximum number of phonons are removed at 6 kOe and thereafter the number decreases causing the thermal resistivity to increase first, pass through a maximum at 6 kOe and then decrease. As the field is increased beyond 12 kOe, there are very few phonons available for scattering and the resistivity approaches its zero field value. For the present system, the spin-phonon coupling constant is found to be 7 = 11.80 cm-l and the line width A = 0.63 cm-l. Since there has been little quantitative work on ErVO,, we are not able to identify the different sources contributing to the line width nor to get an estimate of 7. But it is worth mentioning that they are of the same order as that found by Metcalfe and Rosenberg [4,5] for cerium~ethylsulphate
and holmium ethylsulphate, respectively. The value of the point defect scattering parameter A (A-l = Aw4) is found to be A = 1.42 X 10m46 s4 cm-l which is also of the same order as that found in ref. [4] for CES. The values of the other parameters used in the calculation are u = 5 X lo5 cm/s, the Casimir length L = 0.1384 cm. One of us (VI’S) is indebted to C.S.I.R., India, for financial support. References [l] R. Orbach, Phys. Rev. Lett. 8 (1962) 393. [2] M.J. Metcalfe and H.M.Rosenberg, J. Phys. C5 (1972) 474. [3] J. Callaway, Phys. Rev. 113 (1959) 1046; 122 (1961) 787. [4] P.V.E. McClintock, I.P. Morton, R. Orbachand H.M. Rosenberg, Proc. Roy. Sot. A298 (1967) 359. [5] P.V.E. McClintock and H.M. Rosenberg, Proc. Roy. Sot. A302 (1968) 419.
435