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Transport properties and magnetic phases of NdRu2Si2
r.z
Journal of Maguctism and Magnetic Materials I ! 1 (1992) 239-248 North-! tolland
Transport properties and magnetic phases of NdRu:Siz J.B. Sousa, M.A. Amado, R.P. Pinto, M. Salgueiro Silva, M.E. Braga Centro de Fish'a da Universidade do Porto (INI(') and ;FIMUP, 4000 Porto, Portugal
B. Chevalier and J. E t o u r n e a u Laboratoir; tit" Chimie du Solide du ('NRS. Univ. tit' Bordeatt~ L 351 ('curs &, ia l.d~tYation, 33405 T,dence Cedex. Fran,'e Received 23 October 1991; in revised version 7 January 1992
The tetragonal NdRu:,Si:, compound exhibits a sequence of different magnetic phases below T N = 23.5 K: (i) first, a purely sine-wave modulation with q = (2"rr/aX0.13, 0.12, 0), (ii) below T* -- 15 K this structure gradually squares-up (3 q-harmonics) and (iii) below T¢ --9 K the ferro-magnetic phase sets ip. Accurate measurements of the electrical resistivity (p, d p / d T ) , thermoelectric power {S, d S / d T ) and high resolution differential thermal analysis (DTA) arc reported. Sharp singularities occur in d p / d T and d S / d T at T¢ and T N. The onset o| the square-modulated phase (T < 15 K) produces a pronounced decrease in p(T). The DTA data show that the Ty-anomaly is of 2rid order. Also. for the o r d e r - o r d e r transition at T,. {modulated-lotto) the DTA data give no evidence for a latent heat. A theoretical account is given on the dominant effects observed in each magnetic phase. The anomalous behaviour ol p and S at high temperature are here associated to cryst;~l field effects.
1. Introduction The complex magnetic behaviour of the ternary intcrmctallic c~mpound NdRuzSi 2 (tctragom~l ThCr2Si2-type structure; 14/mmm space groupL has been previously investigated with neutron diffraction and magnetic measurements [1,2]. Below TN = 23.5 K [1] it develops a sinusoidal magnetic modulation with q = (2av/a)(0.13, 0.13, 0), the Nd magnetic moments being parallel to tbe c-axis. A squaring of this st,,ucture occurs below around 15 K (appearance of 3q-harmonics), with an increasing intensity as T decreases. Below T¢----(9 +_ 1) K ferromagnetic order appears with the moments along thc c-axis, but the modulated --* ferromagnetic phase transformation is not complete at 7~.: the neutron resu!ts i:qdi~'ate a Correspondence to: Dr. J.B. Sousa, Centro de F[sica da Universidade do Porto (INIC) and I!FIMUP, Pra~a Gomes Teixeira, 401)0 Porto, Portugal.
rapid reduction of the intensity of the modulated phase as 7" dccrcascs below = 7 K, with the vanishing of the 3rd-ordcr harmonics intensity around 5 K. Below this temperature only the firs~ order term q (sinusoidaily-modulated phase) and the ferromagnetic phase coexist. The,;e features are also reflected in the behaviour of the magnetization [1,3] and of the specific heat [4]. To our knowledge, no detailed investigation has been done on the transport properties of NdRu:Si 2, beyond the simple measurement of the electrical rcsistivity [4]. The work here reported covers high resolution measurements of the electrical resistivity and its temperature derivative (p, dp/dT), the tl-~rmopower coefficients (S, dS/dT) and a detailed thermal-differential-analysis (LYI'A} in NdRu, 51 :. In particular one would like to elucidate: the effect of the squaring of the modulated structure in the electron scattering, the absence of superzope-gap effects, the role played by electron-
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240
J.S. Sousa et al. / Transport properties and magnetic phases of NdRu 2Si2
magnon scattering and crystal field effects, the thermodynamic order of the different magnetic transitions, the role of spin-fluctuations abovc T N and the thermopower magnetic anomalies. Our results show that d p / d T and d S / d T exhibit sharp singularities at T N and T¢ and are very informative on the squaring-up process below 15 K and on the gradual transformation of the modulated phase below T~. An interesting cross-over effect, associated with spin-fluctuations, is observed in d p / d T just above T N. The characteristic superzone-gap humps usually observed in antiferromagnets below T N [5,6] are not observed in our transport properties. Crystal-field effects are observed in the electrical resistivity, and an appropriate separatiot~ has been made. Finally, our DTA data confirms the expected 2nd-order character of the N6el transition, and shows no latent heat at the order-order transition at T¢.
2. Experimental details
2.1. Sample preparation and characterization NdRu2Si 2 was prepared by arc melting of the constituent elements in a purified argon atmosphere and then annealed in an evacuated quartz tube at 800°C for 15 days. The sample was identified by X-ray powder diffraction using a Guinier camera (Cu Kc~ radiation) and their composition and homogeneity was checked by microprobe analysis.
2.2. Electrical resisticity (p, dp / d T ) The electrical resistivity was measured with a four-wire method [7], using a highly stabilized current (I = 100 mA) and a digital nanovoltmeter to measure the voltage across the sample (V). The sample temperature (T) was measured with a Chromel/Au-Feo.07 at% thermocouple giving a resolution of 5 × 10 -4 K. The temperature coefficient of the electrical resistivity ( d o / d T ) was obtained with a technique giving the local values of this derivative within a small temperature interval ( ~ 0.05 K) around each experimental point.
In order to minimize the effects of spurious differential emf's along the leads, the measurements wcrc taken under a quasistatic regime, using a continuous drift of temperature, at a constant rate below 100 mK min -~, in a 3.7 K closed cycle cryostat. An automatic data acquisition system provided (V, T) readings each 2 mK in temperature, and a suitable programme enabled us to obtain accurate values for p and d p / d T at 50 mK intervals.
2.3. Thermoelectric power An automatic dynamic method has been used to measure continuously the thermopower S and its temperature derivative d S / d T from 4 to 300 K. Two copper/chromel junctions directly spotwelded to the sample, were used to measure the small voltages AV~ (copper leads) and AV2 (chromel leads) along the sample when a temperature difference AT was produced with a heater at olle end. A chromcl/Au-Fe0.07 at% thermocouple at the cold end enabled the determination of the sample mean temperature. A high resolution digital source enabled AT to be changed smooth and continuously from 0 to ATmax = 0.5 K, e.g. over a period ot 1 min, during which many simultaneous readings At' t (i), 2~'2 (i) wcre taken with nanovolt resolution. For small ~T, and due to the virtual absencc of thermal lags among copper, chromel and sample at the miniature spotwelded junctions, a plot of Ar,~ (i) versus At: 2 (i) gives a straight line. From the corresponding slope (c~) the thermopowe~" of the sample relatively (e.g.) to copper can be obtained
[s]:
S ( sample / copper) = So( chromel / copper) x c~/(1-ce),
(1)
where So is known at each temperature from a previous calibration of a chromel-copper thermocouple. For each sample mean temperature, S values were obtained both from the c~-slopcs with increasing (0 -* AT.,,x) and decreasing &T (ATm~X 0). The temperature was changed from 4 to 300 K using a closed cycle helium refrigerator with an automatic temperature control, enabling
J.S. Sousa ct al. / Transport pr¢qwrties and magnetw phases of NdRu ,Si,
fairly low constant drifts to span that temperature range ( d T / d t <_ 100 m K . r a i n - ' ) . The data were automatically processed and recorded using a Hewlett Packard 9835A system.
2.4. Diffelential thermal analysis An automatic DTA unit has been locally constructed as a (variant) of the method described by Djurek [9] for the study of magnetic phase transitions. Each small sample (m --- 10-~ g) was suspended by a silk thin thread ia the middle of a cylindrical axial hole in a thick copper block, inside a vacuum chamber (10- ~' Torr). The copper block temperature could be continuously changed from 4 to 300 K at fairly small constant rates (eog. between 20 and 100 m K . m i n - l ) , and a thin chromel/Au-Fe0.07 at% differential thermocouple (O < 0.03 mm) enabled us to measure continuously the small temperature difference 8T between the sample and the copper block. A second thermocouple enabled us to measure the copper block temperature (T), calculating the corrcsponding rate d T / d t at each temperature with a microprocessor. As 6T is very" small ( = 10 inK; duc to the smallness of d T / d t ) all the heat flows between the sample and the copper block are linearizcd in 6T: radiation losses cxAT38T and conduction losses at/38T. The sample heat capacit7 is then given by the approximate expression:
241
3. Experimental results
3. i. Electrical resistitity meaaurcments (p, dp /di) The electrical resistivity (p) and its derivative dp/d7" have been measured between 4 and 300 K. 'I hc residual resistivity of our NdRu2Si 2 sample, P0 = 9 ~1"l. cm, indicates a good quality specimen withcut significant internal microcracks. At high temperatures, the resistivity increases abnost linearly with temperature (see below) as in normal metals, reaching a moderate value of 110 g l l - cm at room temperature. The resistivity slope it then = 0.325 l~fl" cm" K - ' The magnetic transitions of N d R u , S i , [1.2] are clearly marked by the two kinks observed in p(T) as shown in fig. 1. One kink occurs at the N~ei temperature T N = 23.5 K, marking the onset of the incommensurate sinusoidally-modulated phase with wave-vector q = ( w / a ) (0.13, 0.13, 0) [1,2]. The other anomaly occurs in our particular
]
/
TN
k
/ b)
!
20
I
4;
T(tK)
aT3+b
C(T) - - - g T ,
(2)
dT/dt
i
80 where a = A/m and b = p / m . The method is particularly suited to study phase transitions, when interest relies in physical effects within a small temperature range where a K.Otl
qI~LI I K.I.
I.
U~,..
I1. ¢,-1~~'K 9~,¢ 1 |
go
I~.~%J 1 1 0
L f-[ I I L O,
It
TTV
l~Ii i*~.~ v v
L'OIT_ TN
a)
C(T) just outside the critical region (background specific heat, e.g. due to phonons, a simple plot of C ( d T / d t ) / g T versus T 3 gives a and b:
C(dT/dt) &T
= b + a T 3.
(3)
-
010 . . . . . . . ~
i
8 t0
t
16'0
~ T( K ) t
,~
Fig. I. (a) Temperature dependence of the electrical resistMty in NdRu2Si 2. (b) Temperature dependence of the electrical resistivity after deducing the residual resistMt ,'.
J.S. Sousa et al. / Transport properties and magnetic phases of NdRu eSi 2
242
sample at T,. = (8 + 0.5) K, marking the onset of the ferromagnetic phase [1]. The behaviour of dp/dT gives a better insight into these magnetic transitions, as shown in fig. 2. Two sharp positive peaks occur in dp/dT at T N and To. This sensitive transport coefficient exhibits a small but systematic decrease in the paramagnetic phase up to temperatures considerably above T N, due to crystal field effects (see below)• One should also notice the sharp decrease of dp/dT just at T~, changing into a slowly increasing dp/dT-regime at higher temperatures. At low temperatures a sudden flattening-off occurs in the resistivity, leading to dp/dT= 0 below about 5 K (fig. 2).
~N ITc ..
-el
?
J ,b
i
1.0
30
T(K)
__L--
i
.-..~ ~.~..-.'
/ ;"
iS ;
/ I i'
,..-'"'
theory ............. S linear ..............
, s
.........
...'" S
s,S."Sex p
........"""'"' ...'"'" .o
...... ..........•......
a)
i
J
I
360
100
TN
Cr
1
•
"',,, -.5 0
In contrast with the case of normal metals for which S = const × T, the thermopower of NdRu2Si" has an almost constant value of = 4 lxV' K - t from about 80 K up to room temperature (fig. 3). However, for T < 70 K the ther-
tO-
¢; I ,,
T(K ) • ,°
TC I ""'~ "d,;.
b)
;.
,°
""I..,.%,.,
3.2. Thermoelectric power (S)
"7,e-
..:-...
. ~ " ='>'~'~-'.'.:,'-,-,'-.:w .,., ....,........ ~
10
°,
2'o
TCK)
Fig. 3. (a) Temperature dependence of the thermopower in NdRu2Si 2. The dotted line corresponds to phonon and magnetic disorder contributions to the thermopower. The dashed line corresponds to the calculated thermopower considering the crystal-field effects. (b) Temperature dependence of the thermopower in NdRu2Si 2 in an expanded scale.
mopower decreases rapidly, going into negative values at temperatures below = 26 K, then exhibits a minimum around T = 15 K, approaching zero at the lower end of temperature. The magnetic transitions only produce small effects in S(T). As shown in fig. 3b the N6el-point anomaly is associated with a small kink and a subsequent faster decrease of S(T) below T N. Only a faint anomaly i~ observed at the modulated-ferromagnetic transition (To). TlJ'. N~el anomaly produces a very sharp peak in dS/dT, as shown in fig. 4.
3.3. Differential Thermal Analysis (DTA) 0.5 i
i/
0
t
0
100
2OO
3O0
T(K)
Fig. 2. Temperature derivative of the electrical resistivity versus temperature in NdRu2Si 2.
A DTA scan was made over the temperature ~ange 5-30 K. An appropriate data-analysis method [10] enabled us to extract the general behaviour of the specific heat of NdRuzSi 2 (fig. 5), apart from a normalizi~g factor. Finite peaks are observed in C(T), associated with the two magnetic transitions in this compound.
J.S. Sousa et al. / Transport properties and magnetic phases of NdRu zSi :
a peak at Tc, but with extended precursor effects on the high-temperature side (persisting up to about 15 K) and the virtual absence of such effects on the low-temperature side•
~TN
0.2
243
NdR%S; z
4. Data analysis I,•
,,,
° •
°
4.1. Electrical resistit'io, data (p, d p / d T ) •
°
:>
:'.:
~:L i,.-
,
: :;
..'..
:
"0
:':':.
°
• ,
.°
• •
• •
•
•
,
,~
•
i
1
50
•
°
.;.," ,•°
°
I
I
150
T(K)
Fig. 4. Temperature derivative of the thermopower versus
temperature.
The large width of the NEel-point peak indicates that such transition is of second-order. The lower transition (order-order) is associated with
ITN i l'
i
0.6
i •
T..
i
,s
i~J
"
L."
i J
~
;-_-o.~L
u
"
~,
R J
/
-",
J
= /
._~
°
,
/
/
j/
.t
i
/
JI~UUI,,~,,T3
,,'
~o.2~ 0 1"
0.1 . . . . .
10
4.1.1. Basic features The high temperature electrical resistivity slope, d p / d T = 0 . 3 2 5 g f l . c m - K -I, is only slightly higher than in the non-magnetic LaRu :Si 2 compound [11], having d p / d T - - 0.28 ~1'). cm. K-i. This shows that above = 60 K the increase of p in NdRuzSi 2 is essentially dominated by electron-phonon scattering. Below --- 60 K, precursor effects of the antiferromagnetic transition (T N --- 23.5 K) are observed. producing a progressive increase of p above the phonon term. Such initial growth of fluctuations in the paramagnetic phase increases the magnetic electron scattering, producing i;egati~'e d p / d T values (fig. 2). However, as T approaches T N the magnetic electron scattering starts to decrease rapidly, and d p / d T rises sharply to a positive peak at Ty (fig. 2). This crossover near T N, ( T - TN)/T N < O x 10 -2, is likely to be associated with a relatively large conduction electron wavelength Av. If so, only when T is very closed to T N does the coherence length ~(T) of the antiferromagnetic fluctuations (initially smaller than Av) exceed AF and a change in behaviour is expected [12,13]. In commensurate antiferromagnets one usually finds a large negative dip of d p / d T at the N6el point (the onset of antiferromagnetism increases p), due to the magnetic superzone-gap [5,6] which
20
30
T( K )
Fig. 5. Temperature dependence of the normalized specific
heal versus temperature.
e
~II~LI.
IV~
llUlliUt,-,l
UL
lk.UI+4UK,
tlUIII
~,.l+~.-
trons. No such gap effect is observed in NdRu,Si~. This coulo be due to the fact that the particular q-vector of the antiferromagnetic structure in NdRu2Si2 produces gaps out of the Fermi level (for an incommensurate structure one has in fact a set of minigaps spread over the Brillouin zone).
244
J.S. Sous , et aL / Transport properties amt magnetic phases of NdRu 2Sic
Below T N, the increasing magnetic order steadily lowers d p / d T until temperatures around 15 K, where the squaring-up of the sinusoidal phase starts being observed [1]. The progressive squaring-up leads to a rapid decrease of the electron magnetic scattering, originating the increase of d p / d T between 15 K and T~ = 8 K (onset of the ferromagnetic phase in out" data)• Just below ]~ the derivative decreases almost linearly with temperature, but at the lower end of temperature d p / d T suddenly flattens-off around 5 K, virtually vanishing below this temperature (see section 4.1.3).
4.1.2• Separation of the different resisticity contributions
,
4
.~
OL
_
ix
/
t..,,"
I 80
.
I
I 160
I
I
I
260
T(K)
(a) Electron-phonon res;stivity (Pph)"
Previous work on the non-magnetic compounds LaRu2Si 2 [11] and LaNi2Si 2 [14] shows that the electronphonon resistivity behaves normally over the whole temperature range. An available fit of the LaNizSi 2 data [14] to the Block-Griineisen formula gives a Debye temperature O D of 219 K. However, the (constant) high temperature slope ( d p / d T ) . r >> (.~ varies appreciably among RENi2Si 2 compounds, indicating significant changes in the effective number of conducting electrons, thus subtle band overlap effects at the Fermi level [14,15]. This prevents us of taking Pph(T) for NdRuzSi 2 equal to Pph in LaRuzSi 2. We used instead the theoretical BlochGriineisen formula to estimate pph(T) in NdRu2Si 2, taking (9 D = 219 K and fitting the high-temperature slope (dp/dT)T>>o to our experimental value of 0.325 lXf~-cm. Subtracting this Pr,h term (and the residual value p.) from p(T) we obtain the Ap(T)[Ap =p - Po- Pph] curve shown in fig. 6a which contains the still remaining magnetic contributions.
(b) Crystal fieM effects (Pcl). The 2~p versus T curve of fig. 6a reveals the presence of important magnetic contributions clearly showing up at the transition temperatures (TN, T~). However, the curve is rather anomalous above the NEe! temperature: instead of a quick approach to saturation in the paramagnetic phase (total spin-dis-
j/
4-
t
2 0
__.l 0
c)
,
2b
!
'
"
60
'
T(K)
Fig. 6. (a) Temperature dependence of the electrical resistivity after subtracting the residaal resistivity and the phonon resistivity. (b) Temperature dependence of the electrical resistivity after subtracting the residual, phonon and spin disorder resistivity (experiment . . . . and fit . . . . ). (c) Temperature dependence of the magnetic resistivity.
order resistivity; pro=), the resistivity actually increases steadly, almost up to room temperature. The same trend is found if instead of the theoretical Bloch-Griineisen for Pph, we use Pph of LaRu2Si2 multiplied by a suitable constant, to match ( d p / d T ) T >>~.~with our experimental value of 0.325 IXf~" cm. The anomalous slow increase of Ap(T) above T N is here associated with a cry.stal field (CF) effect in NdRueSi 2. As p-saturation occurs only at considerable high temperatures, the relevant CF-level spacing may be rather large in this compound (A > 102 K). We then write: Ap(T) = p , , ( T ) +pd.(T),
(4)
where p,, is the usual spin-disorder resistivity
245
J.S. Sousa et al. / Tra,:~port properties and magnetic phases of NdRu zSi 2
and Pcf represents the crystal field contribution to the electrical resistivity [1,6,7]. Not knowing the exact level spacings in N d R u z S i 2, we use a single spacing (6) crystalfield-model for an order of magnitude estimate [17,18]: Per(T) = pcf(oo) c o s h - 2 ( 6 / 2 k T ) .
(5)
Above -- 70 K, where the spin-disorder resistivity should be constant (Pm~) we can fit the experimental Ap(T) curve to the following expression:
A p ( T ) =pm=+pc,.(~) c o s h - Z ( 6 / 2 k T ) .
(6)
A least squares fit gives Pm~ 6.93 I , O ' c m , pa(m) = 6.83 I.tl]" cm and 6 / k = 226 K. Fig. 6b shows the excellent agreement with the experimental data. =
(c) Spin-disorder resisticity
The good fit obtained at higher temperature with eq. (6), makes reasonable its extrapolation to lower temperatures, enabling a final separation of the spin-disorder resistivity: Pm(T) = A p ( T )
-
where ,-(R;i, T) is the spin-spin correlation function, r =r = r should be valid outside the critical regions. For a modulated phase (q) with the moments along O Z we have:
(Ji>r= Jo'(T) cos( q " Ri)U." .
Inserting this in eq. (8) and using the results &(0)= 1, Y'.j&(Rij)=0 (all j) and Y'-j&(Rq) = - 1 (j 4= i) [19], we obtain: Pro(T) _ - = l + C r 2 ( T ) [ ~
,, o6(R,.)cos(q.R..)]
Pm~
X ~_.,cos2(q'Ri).
(pro).
Pc,(T).
(i) For a ferromagnet we have q = 0 and Y'. cos2(q " Ri)= N, so
0re(T)
- 1 + c r 2 ( T ) Y'~ &(R,,,) = 1 - , r - ' ( T ) .
Pmx
4.1.3. Behat'iour of p(T) in the modulated phases. Squaring-up effects (a) We use the simple expression for the magnetic resistivity in metals with localized moments [12,191: pro(T) Prn~
= 1+
1
Y'~&(Rij)r (Rij, T),
-N ij
(8)
(10)
i
(71
The results are shown in fig. 6c. The extended spin-fluctuatior, effects above T N are now more clearly evidenced than in the original p(T) cnrve. extending up to ~ 70 K. We observe that near and below T N the crystal field effects are negligible. We can also separate the spin-disorder contribution from d p / d T , obtaining the d p , , / d T curve. No significant differences (from d p / c l / ' ) are seen in the cooperative phases, the major changes occurring in the paramagnetic state where the phonon and crystal field play more important roles.
(9)
,71 *
0
(11) The same result holds for an helimagnet described by (J,:) = (1. ( J , ' ) Jcr cos(q" R,) and =
(Ji:'> =J sin(q " Ri). (ii) For a sinusoidal phase ( q 4 = 0 ) w e 2i cos2(q "Ri) = N/2, so
Pro(T)
have
I "~ = 1 + 5o'-(T) Y'~ &(R,,,) cos(q" R,,).
Pm:~
m :# 11
Since &(R,,,) decays rapidly with R,,,, if I qt is small, as it is the case for NdRu2Si 2, the last sum is close to - 1 and then,
p,,,(T)/pm ~= 1-o"
2( T ) / 2 .
(13)
Therefore, in a sinusoidal phase the magnetic ordering (as measured by o-) only produces half the decrease of Pm (in comparison with the cases of ferro and helimagnetism). This is due to the appreciable entropy (then electron-spin disorder
2a,6
J.S. Sousa et al. / Transport properties and magnetic phases of NdRu eSi,
scattering) in the nodal regions of a sinusoidal structure. When we have a succession of sinusoidal and ferromagnetic phases this model indeed predicts the behaviour observed in NdRuzSi 2. (b) In the regions where sin(q.Rj)---0, the thermal average I(Jj) l is very small and spinfluctuations are large. The existance of large local fluctuations invalidates there (nodes) the mean field approximation. Instead we must take into account the effect of local spin-fluctuations, ~Ji = Ji - (Ji), ~Jj = Jj - ( J j ) , which leads to:
<,li "Jy> = (J,> " + ($gi " ~Jy).
(14)
Thus we have an extra resistivity contribution due to (~Ji" ~ J i )" When the modulated structure squares pr,~gressively, the local average I( J i)[ gradually approaches the same (constant)value at all sites and thus local spin-fluctuations are considerably reduced. This explains the faster defirease of p below about 15 K, when the phase squaring progressively takes over, the effect being particularly enhanced as T ~ T~, where ferromagnetic order sets in. In this context we notice that the curvature d2p/dT 2 changes from positive in the sinusoidal phase (q term) towards negative in the squared phase (q plus 3q terms). Recalling that the q and 3q terms in the spontaneous site magnetization grow proportionally to 0.(T) and o-3(T) respectively [20-22], ( J i Z ) / J ~ - c r l ( T ) sin( q . R i )
+ o'3(T ) s i n ( 3 q ' R i ) , (15)
with 0.~ ot ¢r and 0-3 at 0-3, this leads to a resistivity of the form (up to 0-4 terms):
p(T)/pm,~-. 1 - I At 10. 2 - I A310. z.
(16)
Using the crude result o-= ((T N - T)/TN)t 3 with /3 _< i / 2 , one immediately sees that the q term dominates near TN(d2p/dT 2 > 0) whereas the 3q terms dominates further below TN[tr 4 cx(7"N -, )-~t~gives d 2 p / d T 2 < 0, provided/3 > 1/4 which seems physically reasonable]. The faster decrease of p(T) in the squared phase (than predicted for the sinusoidal term) is
also observed in the rare earth Tm [23], which similarly changes from a sinusoidal phase at high temperature to a squared phase at low temperatures (actually not entirely symmetric, but of the ferrimagnetic type). (c) The above considerations may also be relevant to understand the initial enhancement of O produced by spin-fluctuations in the paramagnetic phase (at T>> I'N). The more likely initial short-range order should correspond to "middlelike regions" of a sinusoidai wave (predominantly parallel spins), since in the nodes we have no magnetic order. If the electron wavelength Av is larger than the R E - R E interatomic distances, paired REspins may interact coherently with the same electron wavepacket. In physical terms, a conduction electron then "sees" an effective spin 2J, giving a scattering proportional to 4J2. Fhis represents an enhancement over the incoherent scattering by two independent J-spins, which is proportional to j2 + j2 = 2j2. A simple free-electron calculation of the electron wavelength, A v = 2 r r / K v = (4area2~9) I/3, in fact gives a large value Av --- 6.2 A, if we use n = 3.4 as the number of free electrons per RE-atom [15].
4.2. Thermoelectric power (S, d S / d T ) At high temperatures the thermopower approaches a linear dependence as in normal metals, but below room temperature it exhibits a large positive hump reaching a maximum around 80 K (fig. 3a). The curve closely resembles the behaviour usually observed in systems dominated by crystal field effects [24,25]. This anomalous behaviour is also associated with a crystal field (CF) effect. In this range of temperatures we have: s(r)
= scF(r)
= s0(v),
(17)
where S d is the usual diffusion thermopower having a linear dependence with temperature Sj(T) = 1.23 × 19 -2 T. In order to estimate the Scv contribution we assume a linear extrapolation of the diffusion
,LS. Sousa et al. / Transport properties and magnetic phases of NdRu 2Si 2
thermopower down to lower temperature writting then ScF(T) = S(T) - Sa(T).
(18)
This approximation is not expected to be valid at temperatures far below, but should give the right order of magnitude (and temperature dependence) of S c F ( T ) for T>_ 80 K. We analyse the data using the expression derived for the crystal field contribution, based on the assumption of a single level splitting 6 [24]. SCF ( T ) = const X H ( T/~ ),
(19)
where H(T/6) is a universal function that shows a maximum at T-- 0.3& As shown in fig. 3 the fit to eq. (19) is reasonable when we use 6 = 250 K. In principle the discrepancies could result from the use of a free single electron conduction band, since one knows that S can be very sensitive to the energy derivatives of the density of states at the Fermi level. Below 70 K, S(T) decreases rapidly and reaches negative values below 27 K. This rapid decrease occurs over the same temperature range where the precursor effects of the antiferromagnetic transition show up in dp/dT. The onset of the antiferromagnetic phase (T N = 23.5 K) produces a still faster decrease of S towards more negative values, as shown in the inset of fig. 3. This trend persists down to about 15 K, but below this temperature S(T) rises as T decreases. We associate this new trend with the progressive squaring-up of the modulated antiferromagnetic phase. In contrast to the case of the electrical resistivity, the antiferro-ferromagnetic transition at T--8 K only produces a faint anomaly in S(T), as showr in fig. 3. This simply reflects the intrinsic differences between p and 5' with respect to magnetic scattering. Whereas /9 essentially depends on the mean value of the collision times for electrons with spin up and spin down (~- 1' + ~" $ ), ~he thermopower depends directly on the difference (r $ - r $ ). Thus all the magnetic scattering mechanisms contribute to p, but only..very selective ones contribute to S - just the ones which
247
make r 1' 4: r $ (e.g. random magnetic scattering does not contribute to S). The change of sign of S(T) near the N6el temperature followed by an extremum below it (minimum in our case) resembles the behaviour previously observed in the thermopower of TInS
[81. 4.3. Differential thermal analysis The DTA-data giving the C versus T curve in fig. 5 shows no evidence of a latent heat at the antiferro-ferromagnetic transition (T¢). This is consistent with the neutron diffraction data [1] which shows that the square-modulated phase does ,Jot disappear abruptly at Tc, although being redu"ed rapidly as T decreases in the ferromagnetic phase. Such phase coexistance at and below Tc rules out the possibility of a sharp lst-order transition at Tc. A qualitatively similar C(T) curve has been obtained by Slaski et al. [4], although exhibiting much broader transitions than observed in our sample. In both cases, C(T) exhibits a sharp (almost linear) decrease on the left side of Tc. On the other hand the tail in C(T) extending above Tc up to about 15 K, is a clear indication of the squaring-up of the modulated phase in that temperature range.
References [1] B. Chevalier, J. l~tourneau, P. Hagenmuller, S. Quezei and J. Rossat-Mignod, J. Less-Common Met. 111 (1985) 161. [2] J. Leciejewicz, J. Ptasiewicz-Bak, A. Szytula and M. Slaski, Physica B 130 (1985) 382. [3] I. Felner and I. Nowik, J. Phys. Chem. Solids 45 (1984) 419. [4] M. Slaski, J. Kurzyk, A. Szytuta, B. Dunlap, Z. Sungaila and A. Umezawa, J. de Phys. C 8 (1988) 427. [ql A.R. Markintneh Phys. Re,, I o . o ~loe,~ on [6] R.J. Elliot and F.A. Wedgwood, Proc. Phys. Soc. (London) 81 (1963) 846. [7] R.P. Pinto, M.M. Amado, M.E. Braga, JB. St)usa, P. Morin and R. Ai~onard, J. Magn. Magn. Mater. 72 (1988) 152. [8] A. Amato and J. Sierro, J. Magn. Magn. Mater. 47&48 (1985) 475. [9] D. Djurek and S. Tomic, Phys. Lett. A 85 (1981) 155. t ~
s
. . . . . . . . . . . . . .
,
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . . . .
248
J.S. Sousa et al. / Transport properties arrd magnetic phases o# NdRu 2Si2
[10] R.P. Pinto, M.E. Braga, Poster session presented at the 10th General Conf. of the ('ondcnsed Matter Division, April 1990, Lisboa. [11] M.J. Besnus, J.P. Kappler, P. Lehmann and A. Meyer, Solid State Commun. 55 (1985) 779. [12] S. Alexander, J.S, Helman and J. Balberg, Phys. Rev. B 13 (1985) 304. [13] J.B. Sousa, M.M. Amado, R.P. Pinto, J.M. Moreira and D. Hukin, Commun. Phys. 2 (1977) 95. [14] J.M. Barandiarfin, D. Gignoux, D. Schmitt, J.C. G6mez Sal and J. Rodriguez Fernfindez, J. Magn. Magn. Mater. 69 (1987) 61. [15] M. Slaski, A. Szytula, L. Leciejewicz and A. Zygmunt, J. Magn. Magn. Mater. 46 (1984) 114. [16] P.G. de Gennes and J. Friedel, J. Phys. Chem. Solids 4 (1958) 71.
[17] R.J. Elliott, Phys. Rev. 94 (1954) 564. [18] T. van Peski-Tinbergen and A.J. Dekker, Physica 29 (1963) 917. [19] T.G. Richard and D.J.W. Geldart, Phys. Rev. Lett. 30 (1973) 290. [20] T.A. Kaplan, Phys. Rev. 124 (1961)329. [21] R.J. Elliott, Phys. Rev. 124 (1961) 346. [22] C.S.O. Yokoi, M.D. Countinho-Filho and S.R. Salinas, Phys. Rev. B 24 (198 i ) 4047. [23] S. Lebwold, in: Magnetic Properties of Rare Earth Metals, ed. R.J. Eiliott (Plenum Press, New York, 1972) chap. 7. [24] P. Fulde and i. Peschel, Adv. Phys. 21 (1972) 1. [25] E. Umlaup, G. Pepped and A. Meyer, Phys. Rev. Lett. 30 (1973) 1173.
Journal of Maguctism and Magnetic Materials I ! 1 (1992) 239-248 North-! tolland
Transport properties and magnetic phases of NdRu:Siz J.B. Sousa, M.A. Amado, R.P. Pinto, M. Salgueiro Silva, M.E. Braga Centro de Fish'a da Universidade do Porto (INI(') and ;FIMUP, 4000 Porto, Portugal
B. Chevalier and J. E t o u r n e a u Laboratoir; tit" Chimie du Solide du ('NRS. Univ. tit' Bordeatt~ L 351 ('curs &, ia l.d~tYation, 33405 T,dence Cedex. Fran,'e Received 23 October 1991; in revised version 7 January 1992
The tetragonal NdRu:,Si:, compound exhibits a sequence of different magnetic phases below T N = 23.5 K: (i) first, a purely sine-wave modulation with q = (2"rr/aX0.13, 0.12, 0), (ii) below T* -- 15 K this structure gradually squares-up (3 q-harmonics) and (iii) below T¢ --9 K the ferro-magnetic phase sets ip. Accurate measurements of the electrical resistivity (p, d p / d T ) , thermoelectric power {S, d S / d T ) and high resolution differential thermal analysis (DTA) arc reported. Sharp singularities occur in d p / d T and d S / d T at T¢ and T N. The onset o| the square-modulated phase (T < 15 K) produces a pronounced decrease in p(T). The DTA data show that the Ty-anomaly is of 2rid order. Also. for the o r d e r - o r d e r transition at T,. {modulated-lotto) the DTA data give no evidence for a latent heat. A theoretical account is given on the dominant effects observed in each magnetic phase. The anomalous behaviour ol p and S at high temperature are here associated to cryst;~l field effects.
1. Introduction The complex magnetic behaviour of the ternary intcrmctallic c~mpound NdRuzSi 2 (tctragom~l ThCr2Si2-type structure; 14/mmm space groupL has been previously investigated with neutron diffraction and magnetic measurements [1,2]. Below TN = 23.5 K [1] it develops a sinusoidal magnetic modulation with q = (2av/a)(0.13, 0.13, 0), the Nd magnetic moments being parallel to tbe c-axis. A squaring of this st,,ucture occurs below around 15 K (appearance of 3q-harmonics), with an increasing intensity as T decreases. Below T¢----(9 +_ 1) K ferromagnetic order appears with the moments along thc c-axis, but the modulated --* ferromagnetic phase transformation is not complete at 7~.: the neutron resu!ts i:qdi~'ate a Correspondence to: Dr. J.B. Sousa, Centro de F[sica da Universidade do Porto (INIC) and I!FIMUP, Pra~a Gomes Teixeira, 401)0 Porto, Portugal.
rapid reduction of the intensity of the modulated phase as 7" dccrcascs below = 7 K, with the vanishing of the 3rd-ordcr harmonics intensity around 5 K. Below this temperature only the firs~ order term q (sinusoidaily-modulated phase) and the ferromagnetic phase coexist. The,;e features are also reflected in the behaviour of the magnetization [1,3] and of the specific heat [4]. To our knowledge, no detailed investigation has been done on the transport properties of NdRu:Si 2, beyond the simple measurement of the electrical rcsistivity [4]. The work here reported covers high resolution measurements of the electrical resistivity and its temperature derivative (p, dp/dT), the tl-~rmopower coefficients (S, dS/dT) and a detailed thermal-differential-analysis (LYI'A} in NdRu, 51 :. In particular one would like to elucidate: the effect of the squaring of the modulated structure in the electron scattering, the absence of superzope-gap effects, the role played by electron-
0304-8853/92/$05.00 O 1092 - Elsevier Science Publishers B.V. All rights reserved
240
J.S. Sousa et al. / Transport properties and magnetic phases of NdRu 2Si2
magnon scattering and crystal field effects, the thermodynamic order of the different magnetic transitions, the role of spin-fluctuations abovc T N and the thermopower magnetic anomalies. Our results show that d p / d T and d S / d T exhibit sharp singularities at T N and T¢ and are very informative on the squaring-up process below 15 K and on the gradual transformation of the modulated phase below T~. An interesting cross-over effect, associated with spin-fluctuations, is observed in d p / d T just above T N. The characteristic superzone-gap humps usually observed in antiferromagnets below T N [5,6] are not observed in our transport properties. Crystal-field effects are observed in the electrical resistivity, and an appropriate separatiot~ has been made. Finally, our DTA data confirms the expected 2nd-order character of the N6el transition, and shows no latent heat at the order-order transition at T¢.
2. Experimental details
2.1. Sample preparation and characterization NdRu2Si 2 was prepared by arc melting of the constituent elements in a purified argon atmosphere and then annealed in an evacuated quartz tube at 800°C for 15 days. The sample was identified by X-ray powder diffraction using a Guinier camera (Cu Kc~ radiation) and their composition and homogeneity was checked by microprobe analysis.
2.2. Electrical resisticity (p, dp / d T ) The electrical resistivity was measured with a four-wire method [7], using a highly stabilized current (I = 100 mA) and a digital nanovoltmeter to measure the voltage across the sample (V). The sample temperature (T) was measured with a Chromel/Au-Feo.07 at% thermocouple giving a resolution of 5 × 10 -4 K. The temperature coefficient of the electrical resistivity ( d o / d T ) was obtained with a technique giving the local values of this derivative within a small temperature interval ( ~ 0.05 K) around each experimental point.
In order to minimize the effects of spurious differential emf's along the leads, the measurements wcrc taken under a quasistatic regime, using a continuous drift of temperature, at a constant rate below 100 mK min -~, in a 3.7 K closed cycle cryostat. An automatic data acquisition system provided (V, T) readings each 2 mK in temperature, and a suitable programme enabled us to obtain accurate values for p and d p / d T at 50 mK intervals.
2.3. Thermoelectric power An automatic dynamic method has been used to measure continuously the thermopower S and its temperature derivative d S / d T from 4 to 300 K. Two copper/chromel junctions directly spotwelded to the sample, were used to measure the small voltages AV~ (copper leads) and AV2 (chromel leads) along the sample when a temperature difference AT was produced with a heater at olle end. A chromcl/Au-Fe0.07 at% thermocouple at the cold end enabled the determination of the sample mean temperature. A high resolution digital source enabled AT to be changed smooth and continuously from 0 to ATmax = 0.5 K, e.g. over a period ot 1 min, during which many simultaneous readings At' t (i), 2~'2 (i) wcre taken with nanovolt resolution. For small ~T, and due to the virtual absencc of thermal lags among copper, chromel and sample at the miniature spotwelded junctions, a plot of Ar,~ (i) versus At: 2 (i) gives a straight line. From the corresponding slope (c~) the thermopowe~" of the sample relatively (e.g.) to copper can be obtained
[s]:
S ( sample / copper) = So( chromel / copper) x c~/(1-ce),
(1)
where So is known at each temperature from a previous calibration of a chromel-copper thermocouple. For each sample mean temperature, S values were obtained both from the c~-slopcs with increasing (0 -* AT.,,x) and decreasing &T (ATm~X 0). The temperature was changed from 4 to 300 K using a closed cycle helium refrigerator with an automatic temperature control, enabling
J.S. Sousa ct al. / Transport pr¢qwrties and magnetw phases of NdRu ,Si,
fairly low constant drifts to span that temperature range ( d T / d t <_ 100 m K . r a i n - ' ) . The data were automatically processed and recorded using a Hewlett Packard 9835A system.
2.4. Diffelential thermal analysis An automatic DTA unit has been locally constructed as a (variant) of the method described by Djurek [9] for the study of magnetic phase transitions. Each small sample (m --- 10-~ g) was suspended by a silk thin thread ia the middle of a cylindrical axial hole in a thick copper block, inside a vacuum chamber (10- ~' Torr). The copper block temperature could be continuously changed from 4 to 300 K at fairly small constant rates (eog. between 20 and 100 m K . m i n - l ) , and a thin chromel/Au-Fe0.07 at% differential thermocouple (O < 0.03 mm) enabled us to measure continuously the small temperature difference 8T between the sample and the copper block. A second thermocouple enabled us to measure the copper block temperature (T), calculating the corrcsponding rate d T / d t at each temperature with a microprocessor. As 6T is very" small ( = 10 inK; duc to the smallness of d T / d t ) all the heat flows between the sample and the copper block are linearizcd in 6T: radiation losses cxAT38T and conduction losses at/38T. The sample heat capacit7 is then given by the approximate expression:
241
3. Experimental results
3. i. Electrical resistitity meaaurcments (p, dp /di) The electrical resistivity (p) and its derivative dp/d7" have been measured between 4 and 300 K. 'I hc residual resistivity of our NdRu2Si 2 sample, P0 = 9 ~1"l. cm, indicates a good quality specimen withcut significant internal microcracks. At high temperatures, the resistivity increases abnost linearly with temperature (see below) as in normal metals, reaching a moderate value of 110 g l l - cm at room temperature. The resistivity slope it then = 0.325 l~fl" cm" K - ' The magnetic transitions of N d R u , S i , [1.2] are clearly marked by the two kinks observed in p(T) as shown in fig. 1. One kink occurs at the N~ei temperature T N = 23.5 K, marking the onset of the incommensurate sinusoidally-modulated phase with wave-vector q = ( w / a ) (0.13, 0.13, 0) [1,2]. The other anomaly occurs in our particular
]
/
TN
k
/ b)
!
20
I
4;
T(tK)
aT3+b
C(T) - - - g T ,
(2)
dT/dt
i
80 where a = A/m and b = p / m . The method is particularly suited to study phase transitions, when interest relies in physical effects within a small temperature range where a K.Otl
qI~LI I K.I.
I.
U~,..
I1. ¢,-1~~'K 9~,¢ 1 |
go
I~.~%J 1 1 0
L f-[ I I L O,
It
TTV
l~Ii i*~.~ v v
L'OIT_ TN
a)
C(T) just outside the critical region (background specific heat, e.g. due to phonons, a simple plot of C ( d T / d t ) / g T versus T 3 gives a and b:
C(dT/dt) &T
= b + a T 3.
(3)
-
010 . . . . . . . ~
i
8 t0
t
16'0
~ T( K ) t
,~
Fig. I. (a) Temperature dependence of the electrical resistMty in NdRu2Si 2. (b) Temperature dependence of the electrical resistivity after deducing the residual resistMt ,'.
J.S. Sousa et al. / Transport properties and magnetic phases of NdRu eSi 2
242
sample at T,. = (8 + 0.5) K, marking the onset of the ferromagnetic phase [1]. The behaviour of dp/dT gives a better insight into these magnetic transitions, as shown in fig. 2. Two sharp positive peaks occur in dp/dT at T N and To. This sensitive transport coefficient exhibits a small but systematic decrease in the paramagnetic phase up to temperatures considerably above T N, due to crystal field effects (see below)• One should also notice the sharp decrease of dp/dT just at T~, changing into a slowly increasing dp/dT-regime at higher temperatures. At low temperatures a sudden flattening-off occurs in the resistivity, leading to dp/dT= 0 below about 5 K (fig. 2).
~N ITc ..
-el
?
J ,b
i
1.0
30
T(K)
__L--
i
.-..~ ~.~..-.'
/ ;"
iS ;
/ I i'
,..-'"'
theory ............. S linear ..............
, s
.........
...'" S
s,S."Sex p
........"""'"' ...'"'" .o
...... ..........•......
a)
i
J
I
360
100
TN
Cr
1
•
"',,, -.5 0
In contrast with the case of normal metals for which S = const × T, the thermopower of NdRu2Si" has an almost constant value of = 4 lxV' K - t from about 80 K up to room temperature (fig. 3). However, for T < 70 K the ther-
tO-
¢; I ,,
T(K ) • ,°
TC I ""'~ "d,;.
b)
;.
,°
""I..,.%,.,
3.2. Thermoelectric power (S)
"7,e-
..:-...
. ~ " ='>'~'~-'.'.:,'-,-,'-.:w .,., ....,........ ~
10
°,
2'o
TCK)
Fig. 3. (a) Temperature dependence of the thermopower in NdRu2Si 2. The dotted line corresponds to phonon and magnetic disorder contributions to the thermopower. The dashed line corresponds to the calculated thermopower considering the crystal-field effects. (b) Temperature dependence of the thermopower in NdRu2Si 2 in an expanded scale.
mopower decreases rapidly, going into negative values at temperatures below = 26 K, then exhibits a minimum around T = 15 K, approaching zero at the lower end of temperature. The magnetic transitions only produce small effects in S(T). As shown in fig. 3b the N6el-point anomaly is associated with a small kink and a subsequent faster decrease of S(T) below T N. Only a faint anomaly i~ observed at the modulated-ferromagnetic transition (To). TlJ'. N~el anomaly produces a very sharp peak in dS/dT, as shown in fig. 4.
3.3. Differential Thermal Analysis (DTA) 0.5 i
i/
0
t
0
100
2OO
3O0
T(K)
Fig. 2. Temperature derivative of the electrical resistivity versus temperature in NdRu2Si 2.
A DTA scan was made over the temperature ~ange 5-30 K. An appropriate data-analysis method [10] enabled us to extract the general behaviour of the specific heat of NdRuzSi 2 (fig. 5), apart from a normalizi~g factor. Finite peaks are observed in C(T), associated with the two magnetic transitions in this compound.
J.S. Sousa et al. / Transport properties and magnetic phases of NdRu zSi :
a peak at Tc, but with extended precursor effects on the high-temperature side (persisting up to about 15 K) and the virtual absence of such effects on the low-temperature side•
~TN
0.2
243
NdR%S; z
4. Data analysis I,•
,,,
° •
°
4.1. Electrical resistit'io, data (p, d p / d T ) •
°
:>
:'.:
~:L i,.-
,
: :;
..'..
:
"0
:':':.
°
• ,
.°
• •
• •
•
•
,
,~
•
i
1
50
•
°
.;.," ,•°
°
I
I
150
T(K)
Fig. 4. Temperature derivative of the thermopower versus
temperature.
The large width of the NEel-point peak indicates that such transition is of second-order. The lower transition (order-order) is associated with
ITN i l'
i
0.6
i •
T..
i
,s
i~J
"
L."
i J
~
;-_-o.~L
u
"
~,
R J
/
-",
J
= /
._~
°
,
/
/
j/
.t
i
/
JI~UUI,,~,,T3
,,'
~o.2~ 0 1"
0.1 . . . . .
10
4.1.1. Basic features The high temperature electrical resistivity slope, d p / d T = 0 . 3 2 5 g f l . c m - K -I, is only slightly higher than in the non-magnetic LaRu :Si 2 compound [11], having d p / d T - - 0.28 ~1'). cm. K-i. This shows that above = 60 K the increase of p in NdRuzSi 2 is essentially dominated by electron-phonon scattering. Below --- 60 K, precursor effects of the antiferromagnetic transition (T N --- 23.5 K) are observed. producing a progressive increase of p above the phonon term. Such initial growth of fluctuations in the paramagnetic phase increases the magnetic electron scattering, producing i;egati~'e d p / d T values (fig. 2). However, as T approaches T N the magnetic electron scattering starts to decrease rapidly, and d p / d T rises sharply to a positive peak at Ty (fig. 2). This crossover near T N, ( T - TN)/T N < O x 10 -2, is likely to be associated with a relatively large conduction electron wavelength Av. If so, only when T is very closed to T N does the coherence length ~(T) of the antiferromagnetic fluctuations (initially smaller than Av) exceed AF and a change in behaviour is expected [12,13]. In commensurate antiferromagnets one usually finds a large negative dip of d p / d T at the N6el point (the onset of antiferromagnetism increases p), due to the magnetic superzone-gap [5,6] which
20
30
T( K )
Fig. 5. Temperature dependence of the normalized specific
heal versus temperature.
e
~II~LI.
IV~
llUlliUt,-,l
UL
lk.UI+4UK,
tlUIII
~,.l+~.-
trons. No such gap effect is observed in NdRu,Si~. This coulo be due to the fact that the particular q-vector of the antiferromagnetic structure in NdRu2Si2 produces gaps out of the Fermi level (for an incommensurate structure one has in fact a set of minigaps spread over the Brillouin zone).
244
J.S. Sous , et aL / Transport properties amt magnetic phases of NdRu 2Sic
Below T N, the increasing magnetic order steadily lowers d p / d T until temperatures around 15 K, where the squaring-up of the sinusoidal phase starts being observed [1]. The progressive squaring-up leads to a rapid decrease of the electron magnetic scattering, originating the increase of d p / d T between 15 K and T~ = 8 K (onset of the ferromagnetic phase in out" data)• Just below ]~ the derivative decreases almost linearly with temperature, but at the lower end of temperature d p / d T suddenly flattens-off around 5 K, virtually vanishing below this temperature (see section 4.1.3).
4.1.2• Separation of the different resisticity contributions
,
4
.~
OL
_
ix
/
t..,,"
I 80
.
I
I 160
I
I
I
260
T(K)
(a) Electron-phonon res;stivity (Pph)"
Previous work on the non-magnetic compounds LaRu2Si 2 [11] and LaNi2Si 2 [14] shows that the electronphonon resistivity behaves normally over the whole temperature range. An available fit of the LaNizSi 2 data [14] to the Block-Griineisen formula gives a Debye temperature O D of 219 K. However, the (constant) high temperature slope ( d p / d T ) . r >> (.~ varies appreciably among RENi2Si 2 compounds, indicating significant changes in the effective number of conducting electrons, thus subtle band overlap effects at the Fermi level [14,15]. This prevents us of taking Pph(T) for NdRuzSi 2 equal to Pph in LaRuzSi 2. We used instead the theoretical BlochGriineisen formula to estimate pph(T) in NdRu2Si 2, taking (9 D = 219 K and fitting the high-temperature slope (dp/dT)T>>o to our experimental value of 0.325 lXf~-cm. Subtracting this Pr,h term (and the residual value p.) from p(T) we obtain the Ap(T)[Ap =p - Po- Pph] curve shown in fig. 6a which contains the still remaining magnetic contributions.
(b) Crystal fieM effects (Pcl). The 2~p versus T curve of fig. 6a reveals the presence of important magnetic contributions clearly showing up at the transition temperatures (TN, T~). However, the curve is rather anomalous above the NEe! temperature: instead of a quick approach to saturation in the paramagnetic phase (total spin-dis-
j/
4-
t
2 0
__.l 0
c)
,
2b
!
'
"
60
'
T(K)
Fig. 6. (a) Temperature dependence of the electrical resistivity after subtracting the residaal resistivity and the phonon resistivity. (b) Temperature dependence of the electrical resistivity after subtracting the residual, phonon and spin disorder resistivity (experiment . . . . and fit . . . . ). (c) Temperature dependence of the magnetic resistivity.
order resistivity; pro=), the resistivity actually increases steadly, almost up to room temperature. The same trend is found if instead of the theoretical Bloch-Griineisen for Pph, we use Pph of LaRu2Si2 multiplied by a suitable constant, to match ( d p / d T ) T >>~.~with our experimental value of 0.325 IXf~" cm. The anomalous slow increase of Ap(T) above T N is here associated with a cry.stal field (CF) effect in NdRueSi 2. As p-saturation occurs only at considerable high temperatures, the relevant CF-level spacing may be rather large in this compound (A > 102 K). We then write: Ap(T) = p , , ( T ) +pd.(T),
(4)
where p,, is the usual spin-disorder resistivity
245
J.S. Sousa et al. / Tra,:~port properties and magnetic phases of NdRu zSi 2
and Pcf represents the crystal field contribution to the electrical resistivity [1,6,7]. Not knowing the exact level spacings in N d R u z S i 2, we use a single spacing (6) crystalfield-model for an order of magnitude estimate [17,18]: Per(T) = pcf(oo) c o s h - 2 ( 6 / 2 k T ) .
(5)
Above -- 70 K, where the spin-disorder resistivity should be constant (Pm~) we can fit the experimental Ap(T) curve to the following expression:
A p ( T ) =pm=+pc,.(~) c o s h - Z ( 6 / 2 k T ) .
(6)
A least squares fit gives Pm~ 6.93 I , O ' c m , pa(m) = 6.83 I.tl]" cm and 6 / k = 226 K. Fig. 6b shows the excellent agreement with the experimental data. =
(c) Spin-disorder resisticity
The good fit obtained at higher temperature with eq. (6), makes reasonable its extrapolation to lower temperatures, enabling a final separation of the spin-disorder resistivity: Pm(T) = A p ( T )
-
where ,-(R;i, T) is the spin-spin correlation function, r =
(Ji>r= Jo'(T) cos( q " Ri)U." .
Inserting this in eq. (8) and using the results &(0)= 1, Y'.j&(Rij)=0 (all j) and Y'-j&(Rq) = - 1 (j 4= i) [19], we obtain: Pro(T) _ - = l + C r 2 ( T ) [ ~
,, o6(R,.)cos(q.R..)]
Pm~
X ~_.,cos2(q'Ri).
(pro).
Pc,(T).
(i) For a ferromagnet we have q = 0 and Y'. cos2(q " Ri)= N, so
0re(T)
- 1 + c r 2 ( T ) Y'~ &(R,,,) = 1 - , r - ' ( T ) .
Pmx
4.1.3. Behat'iour of p(T) in the modulated phases. Squaring-up effects (a) We use the simple expression for the magnetic resistivity in metals with localized moments [12,191: pro(T) Prn~
= 1+
1
Y'~&(Rij)r (Rij, T),
-N ij
(8)
(10)
i
(71
The results are shown in fig. 6c. The extended spin-fluctuatior, effects above T N are now more clearly evidenced than in the original p(T) cnrve. extending up to ~ 70 K. We observe that near and below T N the crystal field effects are negligible. We can also separate the spin-disorder contribution from d p / d T , obtaining the d p , , / d T curve. No significant differences (from d p / c l / ' ) are seen in the cooperative phases, the major changes occurring in the paramagnetic state where the phonon and crystal field play more important roles.
(9)
,71 *
0
(11) The same result holds for an helimagnet described by (J,:) = (1. ( J , ' ) Jcr cos(q" R,) and =
(Ji:'> =J sin(q " Ri). (ii) For a sinusoidal phase ( q 4 = 0 ) w e 2i cos2(q "Ri) = N/2, so
Pro(T)
have
I "~ = 1 + 5o'-(T) Y'~ &(R,,,) cos(q" R,,).
Pm:~
m :# 11
Since &(R,,,) decays rapidly with R,,,, if I qt is small, as it is the case for NdRu2Si 2, the last sum is close to - 1 and then,
p,,,(T)/pm ~= 1-o"
2( T ) / 2 .
(13)
Therefore, in a sinusoidal phase the magnetic ordering (as measured by o-) only produces half the decrease of Pm (in comparison with the cases of ferro and helimagnetism). This is due to the appreciable entropy (then electron-spin disorder
2a,6
J.S. Sousa et al. / Transport properties and magnetic phases of NdRu eSi,
scattering) in the nodal regions of a sinusoidal structure. When we have a succession of sinusoidal and ferromagnetic phases this model indeed predicts the behaviour observed in NdRuzSi 2. (b) In the regions where sin(q.Rj)---0, the thermal average I(Jj) l is very small and spinfluctuations are large. The existance of large local fluctuations invalidates there (nodes) the mean field approximation. Instead we must take into account the effect of local spin-fluctuations, ~Ji = Ji - (Ji), ~Jj = Jj - ( J j ) , which leads to:
<,li "Jy> = (J,> "
(14)
Thus we have an extra resistivity contribution due to (~Ji" ~ J i )" When the modulated structure squares pr,~gressively, the local average I( J i)[ gradually approaches the same (constant)value at all sites and thus local spin-fluctuations are considerably reduced. This explains the faster defirease of p below about 15 K, when the phase squaring progressively takes over, the effect being particularly enhanced as T ~ T~, where ferromagnetic order sets in. In this context we notice that the curvature d2p/dT 2 changes from positive in the sinusoidal phase (q term) towards negative in the squared phase (q plus 3q terms). Recalling that the q and 3q terms in the spontaneous site magnetization grow proportionally to 0.(T) and o-3(T) respectively [20-22], ( J i Z ) / J ~ - c r l ( T ) sin( q . R i )
+ o'3(T ) s i n ( 3 q ' R i ) , (15)
with 0.~ ot ¢r and 0-3 at 0-3, this leads to a resistivity of the form (up to 0-4 terms):
p(T)/pm,~-. 1 - I At 10. 2 - I A310. z.
(16)
Using the crude result o-= ((T N - T)/TN)t 3 with /3 _< i / 2 , one immediately sees that the q term dominates near TN(d2p/dT 2 > 0) whereas the 3q terms dominates further below TN[tr 4 cx(7"N -, )-~t~gives d 2 p / d T 2 < 0, provided/3 > 1/4 which seems physically reasonable]. The faster decrease of p(T) in the squared phase (than predicted for the sinusoidal term) is
also observed in the rare earth Tm [23], which similarly changes from a sinusoidal phase at high temperature to a squared phase at low temperatures (actually not entirely symmetric, but of the ferrimagnetic type). (c) The above considerations may also be relevant to understand the initial enhancement of O produced by spin-fluctuations in the paramagnetic phase (at T>> I'N). The more likely initial short-range order should correspond to "middlelike regions" of a sinusoidai wave (predominantly parallel spins), since in the nodes we have no magnetic order. If the electron wavelength Av is larger than the R E - R E interatomic distances, paired REspins may interact coherently with the same electron wavepacket. In physical terms, a conduction electron then "sees" an effective spin 2J, giving a scattering proportional to 4J2. Fhis represents an enhancement over the incoherent scattering by two independent J-spins, which is proportional to j2 + j2 = 2j2. A simple free-electron calculation of the electron wavelength, A v = 2 r r / K v = (4area2~9) I/3, in fact gives a large value Av --- 6.2 A, if we use n = 3.4 as the number of free electrons per RE-atom [15].
4.2. Thermoelectric power (S, d S / d T ) At high temperatures the thermopower approaches a linear dependence as in normal metals, but below room temperature it exhibits a large positive hump reaching a maximum around 80 K (fig. 3a). The curve closely resembles the behaviour usually observed in systems dominated by crystal field effects [24,25]. This anomalous behaviour is also associated with a crystal field (CF) effect. In this range of temperatures we have: s(r)
= scF(r)
= s0(v),
(17)
where S d is the usual diffusion thermopower having a linear dependence with temperature Sj(T) = 1.23 × 19 -2 T. In order to estimate the Scv contribution we assume a linear extrapolation of the diffusion
,LS. Sousa et al. / Transport properties and magnetic phases of NdRu 2Si 2
thermopower down to lower temperature writting then ScF(T) = S(T) - Sa(T).
(18)
This approximation is not expected to be valid at temperatures far below, but should give the right order of magnitude (and temperature dependence) of S c F ( T ) for T>_ 80 K. We analyse the data using the expression derived for the crystal field contribution, based on the assumption of a single level splitting 6 [24]. SCF ( T ) = const X H ( T/~ ),
(19)
where H(T/6) is a universal function that shows a maximum at T-- 0.3& As shown in fig. 3 the fit to eq. (19) is reasonable when we use 6 = 250 K. In principle the discrepancies could result from the use of a free single electron conduction band, since one knows that S can be very sensitive to the energy derivatives of the density of states at the Fermi level. Below 70 K, S(T) decreases rapidly and reaches negative values below 27 K. This rapid decrease occurs over the same temperature range where the precursor effects of the antiferromagnetic transition show up in dp/dT. The onset of the antiferromagnetic phase (T N = 23.5 K) produces a still faster decrease of S towards more negative values, as shown in the inset of fig. 3. This trend persists down to about 15 K, but below this temperature S(T) rises as T decreases. We associate this new trend with the progressive squaring-up of the modulated antiferromagnetic phase. In contrast to the case of the electrical resistivity, the antiferro-ferromagnetic transition at T--8 K only produces a faint anomaly in S(T), as showr in fig. 3. This simply reflects the intrinsic differences between p and 5' with respect to magnetic scattering. Whereas /9 essentially depends on the mean value of the collision times for electrons with spin up and spin down (~- 1' + ~" $ ), ~he thermopower depends directly on the difference (r $ - r $ ). Thus all the magnetic scattering mechanisms contribute to p, but only..very selective ones contribute to S - just the ones which
247
make r 1' 4: r $ (e.g. random magnetic scattering does not contribute to S). The change of sign of S(T) near the N6el temperature followed by an extremum below it (minimum in our case) resembles the behaviour previously observed in the thermopower of TInS
[81. 4.3. Differential thermal analysis The DTA-data giving the C versus T curve in fig. 5 shows no evidence of a latent heat at the antiferro-ferromagnetic transition (T¢). This is consistent with the neutron diffraction data [1] which shows that the square-modulated phase does ,Jot disappear abruptly at Tc, although being redu"ed rapidly as T decreases in the ferromagnetic phase. Such phase coexistance at and below Tc rules out the possibility of a sharp lst-order transition at Tc. A qualitatively similar C(T) curve has been obtained by Slaski et al. [4], although exhibiting much broader transitions than observed in our sample. In both cases, C(T) exhibits a sharp (almost linear) decrease on the left side of Tc. On the other hand the tail in C(T) extending above Tc up to about 15 K, is a clear indication of the squaring-up of the modulated phase in that temperature range.
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J.S. Sousa et al. / Transport properties arrd magnetic phases o# NdRu 2Si2
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