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The temperature dependence of the resistance and the spin density wave energy gap of antiferromagnetic chromium
Solid State Communications,
Pergamon Press
Vol. 13, pp. 1737-1739,1973.
THE TEMPERATURE DEPENDENCE OF THE RESISTANCE WAVE ENERGY GAP OF ANTIFERROMAGNETIC
Printed in Great Britain
AND THE SPIN DENSITY CHROMIUM
Chikahide Akiba and Tadayasu Mitsui Department
of Physics, Faculty of Science, Hokkaido University,
Sapporo, Japan
(Received 10 September 1973 by T. Mzgamiyu)
The anisotropic electrical resistance of antiferromagnetic chromium was measured in the temperature range of 10 K - 200 K. The two-band model was used to analyze the temperature dependence of the resistance, considering the effective number of conduction electrons associated with the SDW energy gap.
A SPIN density wave (SDW) of chromium is formed by itinerant electrons. At the spin-flip temperature, Tr, the polarization of the SDW changes from the direction parallel to the SDW vector Q to the one perpendicular to it. Near T,f( 121 K), an anomaly of the electrical resistivity has been reported by some authors.lm3 However, it was not made clear whether the change of the inclination near T,f of the resistivity versus temperature, in logarithmic scale, was due to the different SDW states. Kashida et al.* have claimed that the change is mainly governed by the mterband scattering of conduction electrons. Meaden et al 3 have reported discontinuous behaviour of the resistivity at 7$. The present experiment was performed in order to clarify the behaviour near T,?of the anisotropic resistance and to study the contribution of the temperature dependence of the SDW energy gap to the conduction electrons.
XL 4.0.
.
.*....
3.5 . .. 3 0-
‘.-.. ‘.A
2.5-
-L Oli
2.01
“...., --.
-..-._ e..... .“..“.
1.5b
50
100
150 200 Tempuratmo ( K I
FIG. 1, xl s i3 In (RI - RL,,)/a In T as a function temperature.
of
at the lowest temperature. Measurements were done after field-cooling processes using a superconductive solenoid, for a parallel state H, II i and a perpendiculal state H, 1 i, where H, is the cooling field of 47.5 kC and i the electric current direction.
The sample of a single crystal was the same as that used in a previous work 5 Measurements of the resistance were done with the Honeywell potentiometer system described in the reference 5. and it was possible to measure reliably 5 0.01 FV. The resistance was measured from 10 K to 200 K, and the measuring voltage varied from cc. 0.2 PV to 130 pV. A Cu-constantan thermocouple was used to determine the temperature of the sample. The thermocouple was calibrated at 4.2 K and 77.4 K, and it was sufficient to obtain an accuracy of 0.1 K
The variation of x E a In (R -&)/a In T with temperature for the perpendicular state is shown in Fig. 1. A least-squares computer fit of five points by a parabola was used to calculate the derivative. x has a maximum at about 50 K, and it should be noticed that above this temperature x decreases nearly continuously through Tsp This behaviour of x is different from the result by Arajs* but similar to the result by Muir et aL6 For the parallel state, the behaviour of x is almost similar to that for the perpendicular 1737
1738
ANTIFERROMAGNETIC CHROMIUM
state. For both states we could not observe any particular anomalies associated with T,t, such as the resistivity-discontinuity observed by Meaden et al. a Our results were analyzed by the two-band model to understand such continuous behaviour of x. The analysis was done based on following assumptions: (1) The conductivity can be written as u = u,, + u,, where cc, is the contribution from the ‘paramagnetic’ sheet of the Fermi surfaces and u. the contribution from the ‘antiferromagnetic’ sheet. (2) The SDW energy gap affects mainly the effective number of conduction electrons for cr.. (3) Only one type of scattering process exists for each of a, and 4, that is, the relaxation times behave as Q a r” and 7. a Tp, respectively, where Q and 0 are non-zero integers. Practically (Yand 0 are 2,3 or 5. Up to a quarter of the Niel temperature TN the magnitude of the SDW energy gap can be considered not to depend on temperature. Therefore, below 75 K the effective number of electron for Us is constant. Below 40 K, the temperature dependence of the resistance seems to be mainly due to Umklappprocesses. Then, with our data between 45 K and 75 K, we can determine the temperature dependences of rP and TV.In this temperature range we find that u is expressed reasonably with the- following expression for both the parallel and perpendicular states:
7
where u is the intrinsic electrical conductivity which is approximated by (R - Ro)-' . Respective values of the residual resistance R,-,were used for R ,, and RI. In Fig. 2, T’/(R - &) vs. TZ is shown for both states. The coefficients are as follows: ,410/A,,o = 1.182 0.02,
&,,/&,
= 1.67 20.05,
B,,e/A,,e = (1.49 f 0.09) x 103. Suffices I/ and 1 denote the parallel and perpendicular states, respectively. Since we did not measure the resistivity, the absolute values of the coefficients could not be determined. The anisotropy of the conductivity should be mainly due to that of 4. Experimentally, the coefficient & has a larger anisotropy than Ao. Therefore, we can consider that the T5-term corresponds to 4.
70
T (K)
a0
100
?
T’ (K')
x 10'
FIG. 2. T’/(R -&) vs TZ for the parallel state and the perpendicular state. Inset shows a schematic drawing of B as a function of temperature. At higher temperatures, the decrease of the SDW energy gap will cause an increase of the coefficient B of the Ts-term. In other words, if we neglect the temperature variation of the electron effective mass, the increase of the effective number of conduction electrons will be reflected in the temperature dependence of B. Assuming that the temperature dependences of the relaxation times are valid up to TN , we can estimate B at TN with previous data near TN.5 And above TN, B is considered to be constant: B,,dB,,t, = (2.92 + 0.15) X IO’, B&Blo
A0 +Bo ‘=p Ts
sp
2t,40, X10'
Vol. 13, No. 10
= (1.68 f 0.05) X 10’.
where $‘s are B’s at TN. In order to see the physical meaning of the temperature dependences of B,, and B,, we define A as follows: A E
[B,-BVJI/[B,--&I,
where B(0
= uaTS = [(R -Ro)
-AoT3]T5.
A,, and AL vs T/TN are shown in Fig. 3. A,, and AL agree well with an energy gap parameter due to the BCS theory, Aecs, shown by solid lines in the figure Thus, we obtain an expression for the conductivity as follows: a=-+
A0
T3
& + (1 - f&s)@,
TS
- Bo) .
It is noted that this expression does not contain any adjustable parameter.s, that is, all the coefficients are definitely determined by observed resistance data only.
Vol. 13, No. 10
ANTIFERROMAGNETIC CHROMIUM
1739
electrons is consistent with the results of the previous work,5 in which we concluded that the contribution to the critical exponent of dR/dT just below TN was the reduction of the conduction electrons proportional to the SDW energy gap.
RG. 3.4, and AL vs reduced temperature. Al, and AL are defmed in the text. Solid lines show an energy gap parameter due to the BCS theory. In antiferromagnetic metals with localized moments, it is known theoretically that the reduction of conduction electrons is proportional to the mag nitude of the energy gap associated with the antiferromagnetic ordering.’ Our experimental results also show for the itineran? SDW chromium that the reduction of the conduction electrons on the antiferromagnetic sheet of Fermi surfaces behaves in the same manner as in the case of the localized antiferromagnet and is just proportional to Anoa. This temperature dependence of the number of conduction
A main modification of the Fermi surfaces due to the SDW is the nesting of the electron- and hole-octahedra, which are considered to be the antiferromagnetic sheet. The anisotropy of the resistance of chromium is mainly due to the difference of Bl10 and &, which originates from the SDW energy gap on the octahedra. And experimentally, it seems that the SDW energy gap does not affect g except for the smalI anisotropy of Ae. The relaxation times of conduction electrons on the paramagnetic sheet, however, might be affected by the existence of the SDW energy gap on the antiferromagnetic sheet. Therefore, although rp and 7, were analyzed in conventional forms, there remain questions as to whether they are simply due to interband electron-phonon scattering and intraband electron-phonon scattering, respectively. B(r) up to TN and the variation of the anisotropy of the resistivity with temperature will be reported and discussed in detail elsewhere.
REFERENCES 1.
MATSUMOTO T., SAMBONGI T. and MITSUI T., J. Phys. Sot. Japan 26,209 (1969).
2.
ARAJS S.,Phys. Lerr. 29A, 211 (1969).
3.
MEADEN G.T. and SZE N.H., Phys. Letr. 30A, 294 (1969).
4.
KASHIDA S., TSUNODA Y. and KUNITOMI N.,J. Phys. Sot. Japan 28.261 (1970).
5.
AKIBA C. and MITSUI T., J. Phys. Sot. Japan 32,644 (1972).
6.
MUIR W.B. and STROM-OLSEN J.O., Phys. Rev. EM, 988 (1971).
7.
MIWA H., Prog. 77reor. Phys. 29,477 (1963).
Der anisotropische elektrische Widerstand des antiferromagnetischen Chroms wurde im Temperaturbereich 10 K < T < 200 K untersucht. Das Zweiband-Model1 wurde gebraucht, urn die Temperaturabhlgigkeit des Widerstands zu analysieren, mit Betracht der effektiven Zahl der Leitungselektronon, die von der SDW-Energielucke herriihren.
Pergamon Press
Vol. 13, pp. 1737-1739,1973.
THE TEMPERATURE DEPENDENCE OF THE RESISTANCE WAVE ENERGY GAP OF ANTIFERROMAGNETIC
Printed in Great Britain
AND THE SPIN DENSITY CHROMIUM
Chikahide Akiba and Tadayasu Mitsui Department
of Physics, Faculty of Science, Hokkaido University,
Sapporo, Japan
(Received 10 September 1973 by T. Mzgamiyu)
The anisotropic electrical resistance of antiferromagnetic chromium was measured in the temperature range of 10 K - 200 K. The two-band model was used to analyze the temperature dependence of the resistance, considering the effective number of conduction electrons associated with the SDW energy gap.
A SPIN density wave (SDW) of chromium is formed by itinerant electrons. At the spin-flip temperature, Tr, the polarization of the SDW changes from the direction parallel to the SDW vector Q to the one perpendicular to it. Near T,f( 121 K), an anomaly of the electrical resistivity has been reported by some authors.lm3 However, it was not made clear whether the change of the inclination near T,f of the resistivity versus temperature, in logarithmic scale, was due to the different SDW states. Kashida et al.* have claimed that the change is mainly governed by the mterband scattering of conduction electrons. Meaden et al 3 have reported discontinuous behaviour of the resistivity at 7$. The present experiment was performed in order to clarify the behaviour near T,?of the anisotropic resistance and to study the contribution of the temperature dependence of the SDW energy gap to the conduction electrons.
XL 4.0.
.
.*....
3.5 . .. 3 0-
‘.-.. ‘.A
2.5-
-L Oli
2.01
“...., --.
-..-._ e..... .“..“.
1.5b
50
100
150 200 Tempuratmo ( K I
FIG. 1, xl s i3 In (RI - RL,,)/a In T as a function temperature.
of
at the lowest temperature. Measurements were done after field-cooling processes using a superconductive solenoid, for a parallel state H, II i and a perpendiculal state H, 1 i, where H, is the cooling field of 47.5 kC and i the electric current direction.
The sample of a single crystal was the same as that used in a previous work 5 Measurements of the resistance were done with the Honeywell potentiometer system described in the reference 5. and it was possible to measure reliably 5 0.01 FV. The resistance was measured from 10 K to 200 K, and the measuring voltage varied from cc. 0.2 PV to 130 pV. A Cu-constantan thermocouple was used to determine the temperature of the sample. The thermocouple was calibrated at 4.2 K and 77.4 K, and it was sufficient to obtain an accuracy of 0.1 K
The variation of x E a In (R -&)/a In T with temperature for the perpendicular state is shown in Fig. 1. A least-squares computer fit of five points by a parabola was used to calculate the derivative. x has a maximum at about 50 K, and it should be noticed that above this temperature x decreases nearly continuously through Tsp This behaviour of x is different from the result by Arajs* but similar to the result by Muir et aL6 For the parallel state, the behaviour of x is almost similar to that for the perpendicular 1737
1738
ANTIFERROMAGNETIC CHROMIUM
state. For both states we could not observe any particular anomalies associated with T,t, such as the resistivity-discontinuity observed by Meaden et al. a Our results were analyzed by the two-band model to understand such continuous behaviour of x. The analysis was done based on following assumptions: (1) The conductivity can be written as u = u,, + u,, where cc, is the contribution from the ‘paramagnetic’ sheet of the Fermi surfaces and u. the contribution from the ‘antiferromagnetic’ sheet. (2) The SDW energy gap affects mainly the effective number of conduction electrons for cr.. (3) Only one type of scattering process exists for each of a, and 4, that is, the relaxation times behave as Q a r” and 7. a Tp, respectively, where Q and 0 are non-zero integers. Practically (Yand 0 are 2,3 or 5. Up to a quarter of the Niel temperature TN the magnitude of the SDW energy gap can be considered not to depend on temperature. Therefore, below 75 K the effective number of electron for Us is constant. Below 40 K, the temperature dependence of the resistance seems to be mainly due to Umklappprocesses. Then, with our data between 45 K and 75 K, we can determine the temperature dependences of rP and TV.In this temperature range we find that u is expressed reasonably with the- following expression for both the parallel and perpendicular states:
7
where u is the intrinsic electrical conductivity which is approximated by (R - Ro)-' . Respective values of the residual resistance R,-,were used for R ,, and RI. In Fig. 2, T’/(R - &) vs. TZ is shown for both states. The coefficients are as follows: ,410/A,,o = 1.182 0.02,
&,,/&,
= 1.67 20.05,
B,,e/A,,e = (1.49 f 0.09) x 103. Suffices I/ and 1 denote the parallel and perpendicular states, respectively. Since we did not measure the resistivity, the absolute values of the coefficients could not be determined. The anisotropy of the conductivity should be mainly due to that of 4. Experimentally, the coefficient & has a larger anisotropy than Ao. Therefore, we can consider that the T5-term corresponds to 4.
70
T (K)
a0
100
?
T’ (K')
x 10'
FIG. 2. T’/(R -&) vs TZ for the parallel state and the perpendicular state. Inset shows a schematic drawing of B as a function of temperature. At higher temperatures, the decrease of the SDW energy gap will cause an increase of the coefficient B of the Ts-term. In other words, if we neglect the temperature variation of the electron effective mass, the increase of the effective number of conduction electrons will be reflected in the temperature dependence of B. Assuming that the temperature dependences of the relaxation times are valid up to TN , we can estimate B at TN with previous data near TN.5 And above TN, B is considered to be constant: B,,dB,,t, = (2.92 + 0.15) X IO’, B&Blo
A0 +Bo ‘=p Ts
sp
2t,40, X10'
Vol. 13, No. 10
= (1.68 f 0.05) X 10’.
where $‘s are B’s at TN. In order to see the physical meaning of the temperature dependences of B,, and B,, we define A as follows: A E
[B,-BVJI/[B,--&I,
where B(0
= uaTS = [(R -Ro)
-AoT3]T5.
A,, and AL vs T/TN are shown in Fig. 3. A,, and AL agree well with an energy gap parameter due to the BCS theory, Aecs, shown by solid lines in the figure Thus, we obtain an expression for the conductivity as follows: a=-+
A0
T3
& + (1 - f&s)@,
TS
- Bo) .
It is noted that this expression does not contain any adjustable parameter.s, that is, all the coefficients are definitely determined by observed resistance data only.
Vol. 13, No. 10
ANTIFERROMAGNETIC CHROMIUM
1739
electrons is consistent with the results of the previous work,5 in which we concluded that the contribution to the critical exponent of dR/dT just below TN was the reduction of the conduction electrons proportional to the SDW energy gap.
RG. 3.4, and AL vs reduced temperature. Al, and AL are defmed in the text. Solid lines show an energy gap parameter due to the BCS theory. In antiferromagnetic metals with localized moments, it is known theoretically that the reduction of conduction electrons is proportional to the mag nitude of the energy gap associated with the antiferromagnetic ordering.’ Our experimental results also show for the itineran? SDW chromium that the reduction of the conduction electrons on the antiferromagnetic sheet of Fermi surfaces behaves in the same manner as in the case of the localized antiferromagnet and is just proportional to Anoa. This temperature dependence of the number of conduction
A main modification of the Fermi surfaces due to the SDW is the nesting of the electron- and hole-octahedra, which are considered to be the antiferromagnetic sheet. The anisotropy of the resistance of chromium is mainly due to the difference of Bl10 and &, which originates from the SDW energy gap on the octahedra. And experimentally, it seems that the SDW energy gap does not affect g except for the smalI anisotropy of Ae. The relaxation times of conduction electrons on the paramagnetic sheet, however, might be affected by the existence of the SDW energy gap on the antiferromagnetic sheet. Therefore, although rp and 7, were analyzed in conventional forms, there remain questions as to whether they are simply due to interband electron-phonon scattering and intraband electron-phonon scattering, respectively. B(r) up to TN and the variation of the anisotropy of the resistivity with temperature will be reported and discussed in detail elsewhere.
REFERENCES 1.
MATSUMOTO T., SAMBONGI T. and MITSUI T., J. Phys. Sot. Japan 26,209 (1969).
2.
ARAJS S.,Phys. Lerr. 29A, 211 (1969).
3.
MEADEN G.T. and SZE N.H., Phys. Letr. 30A, 294 (1969).
4.
KASHIDA S., TSUNODA Y. and KUNITOMI N.,J. Phys. Sot. Japan 28.261 (1970).
5.
AKIBA C. and MITSUI T., J. Phys. Sot. Japan 32,644 (1972).
6.
MUIR W.B. and STROM-OLSEN J.O., Phys. Rev. EM, 988 (1971).
7.
MIWA H., Prog. 77reor. Phys. 29,477 (1963).
Der anisotropische elektrische Widerstand des antiferromagnetischen Chroms wurde im Temperaturbereich 10 K < T < 200 K untersucht. Das Zweiband-Model1 wurde gebraucht, urn die Temperaturabhlgigkeit des Widerstands zu analysieren, mit Betracht der effektiven Zahl der Leitungselektronon, die von der SDW-Energielucke herriihren.