3. THERMAL PROPERTIES UNDER HIGH PRESSURE 1 General survey
3.4 Thermal expansion at high pressure
and thermal expansion measurements under pressure shed light on this problem and revealed that the pseudo gap state is formed below the critical pressurePc2, above which a magnetically ordered state appears.
Figure 38showsCof SmS under pressure. As the temperature is raised, C initially increases linearly with T (see the inset), indicating nonzero electronic specific heat g. Then, a broad peak of heat capacity forms, which can be described by a conventional Schottky model with an energy gap D. The Schottky peak shifts to lower temperatures with increasing pressure, that is,D decreases with P. The presence of the energy gap is also observed in thermal expansion measurements under pressure.
Figure 39shows theTdependence of the linear thermal expansion coeffi- cienta at pressures up to 2.16 GPa. In the ‘‘black phase’’ belowPc1,a is small and shows weakTdependence. When the system transforms into the golden phase, aLsuddenly changes and becomes strongly T dependent with a large negative peak. When pressure exceeds Pc21.9 GPa, the Schottky anomaly disappears, and instead a sharp anomaly with a positive sign appears atTM11 K. As seen inFigure 40, the same anomaly was also detected by the ac specific heat measurements. Together with the observation of the internal field by nuclear forward scattering experiments (Barla et al., 2004), we ascribe the sharp anomaly to the phase transition between the paramagnetic and magnetically ordered states.
same as that of LalnCu2above ca. 150 K, but the difference between them becomes very large as temperature decreases. a100 of CelnCu2 is 9 106 K1 near 10 K, which is larger than that of LalnCu2 by an order of magnitude.
InFigure 43, we plotteda/Tas a function ofT2for CelnCu2(a100) and LaInCu2. The value ofa/Tof CelnCu2is found to have a large increase
1.93
2.16
DL/L
1.73 0.74
aL(K–1)⫻10–4
0.3 GPa 0.3 GPa
ambient
ambient0
–0.5 0 0.5 1 1.5
10
0 20 30 40 50 60
100 200
T (K)
T (K)
2⫻10–3
0⫻100
–2⫻10–3
300
2.05 GPa TM
FIGURE 39 Thermal expansion coefficientaof SmS as a function of temperature at selected pressures (Imura et al., 2008).
SmS
00 2 4 6
10 20 30
T (K)
Cac/T (a.u.)
3.1 GPa 4.0 3.6 4.6
6.0 7.5
8.1
FIGURE 40 Typical specific heat curves of SmS shown as temperature variation of Cac/Tfor several pressures. The data are normalized to 30 K (Haga et al., 2004).
with decreasing temperature below ca. 20 K. On the other hand, thea/Tof LalnCu2is nearly independent of temperature. This behavior is in sharp contrast to that of CelnCu2. The large enhancement in the magnitude of a/Twas observed in other HF materials such as CeA13(Kagayama et al., 1990), CeCu6(Oomi et al., 1990b), CeCu2Si2(Takakura et al., 1990), and
at ambient pressure 0
0 50 100 150
T (K)
Dl/l
200 250 300
1 2 3 4 5
⫻10–3
CeInCu2
LaInCu2
FIGURE 41 Temperature dependence ofDl/lalong [100] of single-crystalline CeInCu2
and also for LaInCu2as a reference (Oomi et al., 1990a).
CeInCu2
LaInCu2
00 10 20
100 a⫻106 (K–1)
200 T (K)
300
FIGURE 42 Thermal expansion coefficientaof CeInCu2and also for LaInCu2as functions of temperature (Oomi et al., 1990a).
UPt3(Visser et al., 1985). These results show that it is one of the character- istic features of HF materials.
The behavior of a/T at low temperatures resembles large enhance- ment ofC/Tin HF materials. Taking into account the Gru¨neisen relation, which shows that a is roughly proportional to specific heat C, a is expected to have the same temperature dependence as C, that is, the present results seem to be consistent with this notion.
Since the a–T curve of LaInCu2 is similar to that of normal metals (White and Collins, 1972), the approximation in which the a(LalnCu2) originates mainly from phonon contribution may be reasonable. Here, we describe the observeda(CelnCu2) as
aðCeInCu2Þ ¼amagþaphffiamagþaðLaInCu2Þ; (26) whereamagis the magnetic contribution toadue to 4f electrons. Thenamag
is approximated by the following equation:
amagffiaðCeInCu2Þ aðLaInCu2Þ: (27)
Figure 44 showsamag as a function ofTfor the [100] direction. amag
increases with increasing temperature until it reaches a maximum around 25 K and then decreases with increasing temperature above 25 K. The origin of the maximum inFigure 44is considered to arise from the CEF splitting of Ce3þion and then expected to behave as Schottky anomaly (Schefzyk et al., 1985).
Finally, we briefly discuss these results on the basis of our phenome- nological theory (Oomi et al., 1988b).amagis shown to have the so-called Schottky-type temperature dependence if we assume a cubic symmetry
CeInCu2 a/T⫻106 (K–2)
00 0.5 1.0 1.5
1000 T2(K2)
2000 LaInCu2
FIGURE 43 Values ofa/Tas functions ofT2for CeInCu2and LaInCu2(Oomi et al., 1990a).
and two levels with an energy difference D (K). amag has a maximum around 0.4D(¼Tmax). From the present data in whichTmaxis 25 K,Dis estimated to be 63 K. This value is in good agreement with D¼65 K obtained byŌnuki et al. (1987). The maximum inamag, (amag)max, is given by the following equation (Oomi et al., 1988b):
amag
max 0:44kkB
3V0 ; (28)
where k is the compressibility, kB is the Boltzmann constant, V0 is the volume without CEF splitting, and is@ln D/@lnV. By substitutingV0
and k for 7.81027 m3 and 0.9102 GPa1¼0.91011 m3/J (Oomi et al., 1988a) and (amag)max 7.6106K1, we obtain ¼30.
Thus, the result means that the value ofDincreases with pressure having a rate of 18 K GPa1, which is about five times larger than that ofTK,@TK/
@P 4 K GPa1(Kagayama et al., 1991a).
Figure 45 shows the temperature dependence of the linear thermal expansion coefficient a of CeInCu2 below 80 K at various pressures (Kagayama et al., 1994d). At 0.3 GPa,adecreases smoothly with decreas- ing temperature but is still large around 10 K: it is about 300 times as large as that of Cu at 10 K. Such behavior is commonly observed in HF com- pounds. With increasing pressure a decreases, which implies that the DOSs decreases at high pressure as will be discussed later.
InFigure 46, the value ofa/Tis plotted as a function ofT2at various pressures. a/T increases rapidly on cooling, likeC/T, implying a large enhancement ofD(eF) or effective mass of electrons at low temperature.
amag=a(CelnCu2)
amag⫻106 (K–1) –a(LaInCu2)
T (K) 00
2 4 6
100 200
FIGURE 44 Magnetic contribution to the thermal expansion coefficient,amag¼ a(CeInCu2)a(LaCuIn2), as a function of temperature (Oomi et al., 1990a).
By applying pressure, the value ofa/Tis suppressed: around 10 K,a/Tat 2 GPa becomes about half of that at ambient pressure. The value ofA0in Eq. (23) is generally dependent on temperature. In this review, it is assumed that a/T at a certain temperature is roughly proportional to the magnitude of A0. Figure 47 shows the pressure dependence of the value ofa/Tat 7 K. The ratioa/Tdecreases by applying pressure. In order
CelnCu2 0.3 GPa
0.7 GPa
2.0 GPa
T (K) a (10–6 K–1)
0 0 5 10 15 20
20 40 60 80
FIGURE 45 The linear thermal expansion coefficientaof CeInCu2as a function of temperature at high pressure (Kagayama et al., 1994d).
00 0.5 1.0
CeInCu2
a/T (10–6 K–2)
0.3 GPa 0.7 GPa 2.0 GPa
1.50 200 400
T2 (K2)
T (K)
600 800
10 20 30
FIGURE 46 The liner thermal expansion coefficientaof CeInCu2as a function of temperature at high pressure (Kagayama et al., 1994d).
to derive the volume dependence ofA0, the Gru¨neisen parameterGA0for A0was estimated by fitting the data to the following equation:
A0¼A00 V V0
GA0
; (29)
whereA00 andV0are the zero pressure values ofA0andV, respectively.
Using X-ray diffraction measurements at high pressure (Kagayama et al., 1990) to determine the relative change of volumeV/V0in Eq.(29),GA0was calculated to be34.
In a rough approximation thatA0is inversely proportional toTK,GA0
corresponds to the Gru¨neisen parameter ofTK,GK @lnTK/@lnV(see Eq.(3)).Kagayama et al. (1992a)mentioned that the Gru¨neisen parameter ofTKfor CeInCu2is about 60, which was obtained by the volume depen- dence of the resistivity-maximum temperature or the coefficient of T2- dependent resistivity. The present value of 34 is of the same order of magnitude.
Finally, we comment about the pressure change in the overall behavior of a. Below 50 K, thea–T curve at 0.3 GPa has a negative curvature in contrast to those of normal metals (White and Collins, 1972). This behav- ior was also observed at ambient pressure but becomes less prominent at high pressure as is seen from Figure 45. The a–T curve of CeInCu2 at 2 GPa seems to be similar to that of LaInCu2. Such behavior at high pressure was observed also for CeNi compound (Okita et al., 1991) at high pressure as will be mentioned inSection 3.4.3. This fact implies that volume contraction shifts the system from the well-localized 4f (smallTK) state into the itinerant (largeTK) state: such pressure-induced crossover in the electronic state has been observed in several other Ce-based HF compounds.
1.5
1.0
0.50 1
P (GPa)
2 CeInCu2
a/T (10–6 K–2)
at 7 K
FIGURE 47 Pressure dependence ofa/Tof CeInCu2at 7 K (Kagayama et al., 1994d).
3.4.1.2 CeCu6 Next we consider CeCu6, which is also a typical example of HF materials having thegvalue of 1.7 J/mol K2(Satoh et al., 1989). The crystal structure is orthorhombic.Figure 48shows the thermal expansion coefficients of single-crystalline CeCu6for each crystal axis. Large anisot- ropy in theai–Tcurves (i¼a,b, andc) is seen in this figure. The tempera- ture dependence ofaais similar to that of a normal metal such as Cu or Ag. On the other hand,abshows anomalous temperature dependence: it increases rapidly with increasing temperature below 10 K, the rate of increase becomes small, showing a broad maximum around 80 K and then abdecreases slowly above 100 K. The value of abat 10 K is about 10l06K1, which is larger than that of Cu (0.03106K1) by two orders of magnitude.acalso shows anomalous temperature dependence:
acdecreases with decreasing temperature, having the same temperature
CeCu6
CeCu6
CeCu6 ai (⫻10–6 K–1 , i=a, b, or c)
LaCu6
LaCu6
LaCu6
T (K) 00
10 0 10 0 10 20
100 200
a-axis
b-axis
c-axis
FIGURE 48 Temperature dependencies of linear thermal expansion coefficients, ai¼(1/li)(dli/dT), of single-crystalline CeCu6(circles) and La (triangles), whereidenotes a,b, andc(Oomi et al., 1990c).
dependence as that of LaCu6until it becomes zero around 30 K, and then it begins to increases upon further cooling to the value of about 7l06 K1 near 5 K. Such increase was also found in the thermal expansion coefficient of single-crystalline UPt3(Visser et al., 1985).
Results obtained for CeCu6are similar to those for CeInCu2, where a large enhancement in the magnitude ofA0, that is, in the electronic and magnetic contributions was observed (Oomi et al., 1990b). The large enhancement in the magnitude of a/T is also observed in other HF materials including U-compounds. Considering the fact that there are no such effects in the normal metals and alloys, the anomalously large enhancement ofa/Tis one of the characteristics of Ce- and U-based HF materials. Further, as was already noted, the anomalous behavior of the thermal expansion coefficient is closely related to the anomalous behavior of heat capacity C by the Gru¨neisen relation, Eq. (21). In Table 3, we TABLE 3 Relation between the values ofa/TandC/Tat 10 K for HF compounds and reference materials (Y and La compounds)
Compound a/T (109K2)
(at 10 K) Reference
C/T(mJ/
mol K2)
(at 10 K) Reference
CeAl3 600 Kagayama
et al. (1990)
350 Flouquet et al.
(1982)
CeCu6 430 Oomi et al.
(1990c)
360 Kato et al. (1987) CeCu2Si2 360 Takakura
et al. (1990)
140 Stewart (1984) CeInCu2 1000 Oomi et al.
(1989)
340 Ōnuki et al. (1987)
CeCu2 30 Uwatoko et al.
(1990)
180 Ōnuki et al.
(1985b) CeRu2Si2 170 Lacerda et al.
(1989)
220 Fisher et al. (1988)
YCu2 10 Uwatoko et al.
(1990)
50 Luong et al. (1985)
LaCu6 5 Oomi et al.
(1990c)
120 Kato et al. (1987)
LaAl3 10 Kagayama
et al. (1990)
21 Edelstein et al.
(1987) LaCu2Si2 40 Takakura
et al. (1990)
33 Takeda et al.
(2008) LaInCu2 240 Oomi et al.
(1989)
166 Sato et al. (1992)
summarized the data of C/T and a/T. It is found that there is a close relation between C/T and a/T of HF compounds. In conclusion, we emphasize that large enhancement ofa/T (oraV/T) at low temperature should be one of the most important characteristics of HF materials.
Next we will consider the effect of pressure on the anisotropic proper- ties in the thermal expansion coefficients of single-crystalline CeCu6. Figure 49showsaaat ambient and 0.82 GPa. No significant difference in theaais found within experimental errors between ambient and 0.82 GPa.
In Section 2, the linear compressibility along a-axis, ka, is very small compared to kb andkc(Shibata et al., 1986). This fact indicates that the binding force along the a-axis is the largest among the three axes. The length of thea-axis is not affected much by the magnetic field (Oomi et al., 1988a). From these facts, thea-axis is not significantly affected by apply- ing external forces.
ab–Tcurves of CeCu6are shown inFigure 50at ambient pressure, 0.76 and 1.26 GPa. The value of ab is found to be affected significantly by applying pressure:abat 50 K is 12106K1at ambient pressure and 8.0106K1at 1.26 GPa. This result is in sharp contrast with that ofaa. In the foregoing paragraph, it was mentioned that there is a large mag- netic contribution to the magnitude ofab, which is due to the existence of unstable 4f electrons. Thus the large effect of pressure on the magnitude ofabis caused by the instability of 4f electronic states in CeCu6.
Temperature dependence of ac of CeCu6 below 50 K is shown in Figure 51 at ambient pressure, 0.32, 0.63, and 1.26 GPa. The overall behavior ofac–Tcurve at high pressure is almost similar to that at ambient pressure. The temperature showing the minimum in ac,Tmin, is almost independent of pressure. But the value of ac(Tmin) is found to increase with increasing pressure from a small negative value to a positive one at
0.82 GPa
0 GPa a-axis
aa (⫻10–6 K–1)
CeCu6 00
10 20
50 100
T (K)
150 200
FIGURE 49 Temperature dependence ofaaof CeCu6at ambient pressure and at 0.82 GPa (Oomi et al., 1991).
0.63 GPa. Sinceaccurve at 0.63 GPa is the same as that at 1.26 GPa,ac(Tmin) is independent of pressure above 0.63 GPa. The origin of this behavior is not clear.
It should be noted that the maximum in ther–Tcurve,rmax, of CeCu6
decreases with increasing pressure, but above 1 GPa,rmaxbecomes nearly constant (Thompson and Fisk, 1985). This was explained by assuming that the valence of Ce atom changes significantly at low pressure but the change becomes small at high pressure. Thermal expansion results may be interpreted on the basis of the same assumption. Further, it is interest- ing to note thatac(Tmin) changes significantly below 1 GPa, but it shows saturation above 1 GPa, which is the same behavior as that ofrmax. This fact suggests that the pressure dependence of ac is dominated by the valence change at high pressure. In the foregoing section, it was reported a large enhancement of the value of a/T at low temperature, which is
00 10 20
50 ab (⫻10–6 K–1)
100 T (K)
1.26 GPa 0.76 GPa 0 GPa
CeCu6 b-axis
150 200
FIGURE 50 Temperature dependence ofabof CeCu6at ambient and high pressures (Oomi et al., 1991).
ac (⫻10–6 K–1) 1.26 GPa 0.63 GPa 0.32 GPa
0 GPa CeCu6 50 40
30 T (K) 20
10 0.0 5.0
c-axis
FIGURE 51 Temperature dependence ofacof CeCu6at high pressure (Oomi et al., 1991).
originated from the enhancement of the effective mass of conduction electrons. As mentioned above, the change in the magnitude of a by applying pressure is dominated by a decrease in the value of ab, which decreases considerably at high pressure. Considering this fact, it is expected that the enhancement ofal/Tat low temperature is suppressed by applying pressure. This means that the effective mass of electrons decreases at high pressure. This conclusion is consistent with the results obtained by heat capacity measurement at high pressure as mentioned in Section 3.3.1.
3.4.2 Kondo compounds with magnetic order
There are numerous materials showing Kondo effect accompanied by magnetic order, antiferrromagnetism or ferromagnetism. In this section, we show the experimental results for antiferromagnetic materials, CeRh2Si2(TN ¼36 K), CeAu2Si2(TN¼8 K), and ferromagnetic material, GdAl2 (TC¼162 K). Since the electrical resistance and other physical properties of these compounds will be mentioned in detail inSection 4, we will confine our discussion here to thermal properties. It is well known that in these compounds, the Kondo effect competes with the Ruderman–
Kittel–Kasuya–Yoshida (RKKY) interactions, which induce magnetic ordering. A variety of magnetic properties of rare earth compounds have been explained on the basis of the interplay of these interactions.
Doniach presented a theoretical phase diagram by taking into account these facts to explain the electronic and magnetic states of rare earth compounds which is the so-called Doniach’s phase diagram. The rough schematic of this diagram is depicted in Figure 52. In the diagram, the characteristic temperatures such as TN, TK, and TRKKY are shown as a function of J/W, where Jis the exchange coupling strength, and band- width W is nearly proportional to 1/D(eF). The magnitude of J/W is
Temperature
Magnetic 4f-metal Magnetic CKS Nonmagnetic CKS / W TRKKY
TM
TK
FIGURE 52 Doniach phase diagram afterBrandt and Moshchalkov (1984).
proportional to pressure. TheTMhas a peak at a value of J/W, where TK¼TRKKY. At present, CeRh2Si2is considered to be on the right hand side of the peak, but CeAu2Si2is on the left hand side.
3.4.2.1 CeRh2Si2 CeRh2Si2 crystallizes in the ThCr2Si2 tetragonal struc- ture and shows two kinds of antiferromagnetic orderings at 36 K (¼TN1) and 24 K (¼TN2) (Kawarazaki et al., 2000). More details about this com- pound will be mentioned in Section 4.2.1. It has been reported that TN1
disappears at a relatively low pressure around 1 GPa, in spite of such high TN(Settai et al., 1997). The resistivity anomaly atTN1becomes broader by applying high pressure, and no anomaly is observed atTN2.The exact value of the critical pressurePc, where the AFM disappears, is difficult to identify from earlier electrical measurements (Ohashi et al., 2002). Each magnetic transition has been reported to be accompanied by pronounced thermal expansion anomalies.
Figure 53(Honda et al., 1999) shows the fractional change in length Dl/l as a function of temperature T below 300 K. As temperature increases, the value of Dl/l decreases gradually followed by two steps near 25 K (¼TN2) and 37 K (¼TN1) due to magnetic phase transitions and then begins to increase smoothly above 38 K. A clear spontaneous MS is observed belowTN1having an order of magnitude of 104.
Figure 54shows the temperature evolution ofDl/lin the range from 10 to 50 K at high pressure below 2.15 GPa. Two distinct anomalies due to magnetic transition are seen in the temperature dependence of Dl/l at ambient pressure as mentioned above. Below 1.1 GPa (Pc), the sponta- neous MS is clearly observed. Since both the MS effect and the thermal
2
⫻10–3
0 GPa
1
0
0 100 200
T (K)
Δl/l
300
FIGURE 53 Linear thermal expansion of CeRh2Si2at ambient pressure (Honda et al., 1999).
expansion anomalies are not detected above 1.1 GPa, that is, the thermal expansion Dl/l (and also the thermal expansion coefficients) increases smoothly with temperature. The critical pressure Pc where the AFM disappears is assumed to be around 1.1 GPa. It is seen that the magnetic phase transition temperatures, TN1 and TN2 indicated by the arrows in Figure 54, decrease with increasing pressure.
An example ofa–Tcurves is shown at various pressures inFigure 55 below 50 K. Significant changes inaare easily seen as pressure increases.
Two anomalies are found at ambient pressure at TN1¼37 K and TN2¼26 K, but at 1.06 GPa, one broad minimum in the a–T curve is observed around 15 K, that is,TN2disappears above this pressure. Above 1.09 GPa, no anomalies are found in the behavior of the thermal expan- sion coefficient. In other words, the AFM is suppressed completely above 1.09 GPa. TN1 and TN2 are found to decrease rapidly with increasing pressure and extrapolated to be 0 around 1.11 GPa (¼Pc1) and 0.55 GPa (¼Pc2), respectively. The pressure dependence of TN1 and TN2 will be summarized inFigure 57. At high pressure around 1.06 GPa,ashows a
TN2
TN1
1.0⫻10–4
0.5 GPa
2.15 GPa
10 20 30
T (K)
40 50
1.11 GPa
Δl/l
0 GPa
FIGURE 54 Linear thermal expansion of CeRh2Si2at low temperature at various pressures. The arrows indicate the magnetic phase transition temperatures,TN1andTN2
(Honda et al., 1999).
broad maximum around 30 K, which may be due to the CEF splitting of the sixfold degenerate 4f levels of Ce3þion. Since the maximum is suppressed with increasing pressure, the present result indicates that the effect of CEF splitting disappears at high pressure. In other words, the 4f electron of Ce is delocalized at high pressure. The magnetic properties of Ce compounds at high pressure are generally explained using the Doniach phase diagram assuming thatJD(eF) is proportional to pressureP.
For CeRh2Si2, the RKKY interactions overcome the Kondo interactions at ambient pressure because the ground state is antiferromagnetic. When pressure increases, the Kondo interactions increase more rapidly than the RKKY interactions. This means that the localized 4f moment disappears at a critical pressure Pc (¼1.1 GPa) inducing a Kondo spin compensated state that has large critical fluctuations. Just abovePc, we expect a large contribution of spin fluctuations to physical quantities, which manifests itself as large values of a and of the coefficient of the T2 term in the electrical resistivity. The value of a around 10 K is 2–3106 K1 at 1.11 GPa, which is much larger than that of normal metals such as Cu, a(10 K)¼0.03 106K1. Considering thatTNdecreases with pressure, CeRh2Si2should be placed on the right hand side of the peak of the phase diagram and then the Kondo state is stabilized above a critical value of JorPc.
10
0
–10
–200 1.06 GPa a (K–1)
1.09 GPa
⫻10–6
0.95 GPa CeRh2Si2
1.5 GPa
0 GPa
10 20 30
T (K)
40 50
FIGURE 55 Thermal expansion coefficientsaof CeRh2Si2at high pressure (Oomi and Kagayama, 2006).